Clémentine Garric

Aïda Sebaa

Florent Caetta

Céline Perez

Julien Savatovsky

Claire Sergent

Sylvie Chokron
email: sylvie.chokron@gmail.com

Dissociation between objective and subjective perceptual experiences in a population of hemianopic patients: A new form of blindsight?

Keywords: blindsight, hemianopia, post-chiasmatic lesion, blind visual field, awareness

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Introduction

The most common visual defect induced by a unilateral lesion of the retrochiasmal pathways is homonymous hemianopia (HH) [START_REF] Cavézian | Specific impairments in visual processing following lesion side in hemianopic patients[END_REF][START_REF] Zhang | Homonymous hemianopias: Clinical-anatomic correlations in 904 cases[END_REF], whereby patients are blind in the contralesional visual field of each eye. Hemianopic patients can exhibit implicit residual capacities in the so-called "blind" visual field, despite exhibiting a severe contralesional visual-field defect. Such residual capacities are usually referred to as blindsight, a term coined by [START_REF] Sanders | BLINDSIGHT": VISION IN A FIELD DEFECT[END_REF] that has inspired debate [START_REF] Cowey | The blindsight saga[END_REF][START_REF] Overgaard | Reconciling current approaches to blindsight[END_REF][START_REF] Sanders | BLINDSIGHT": VISION IN A FIELD DEFECT[END_REF][START_REF] Stoerig | Blindsight, conscious vision, and the role of primary visual cortex[END_REF][START_REF] Weiskrantz | Review Lecture: Behavioural analysis of the monkey's visual nervous system[END_REF].

Blindsight was first described in patients who could accurately steer their gaze toward a flashing light presented in their blind field, yet who did not report having perceived the stimulation [START_REF] Poppel | Residual Visual Function after Brain Wounds involving the Central Visual Pathways in Man[END_REF]. Another blindsight patient, G.Y., reported awareness of certain stimuli despite not attesting to any concrete visual perception of them [START_REF] Weiskrantz | Parameters affecting conscious versus unconscious visual discrimination with damage to the visual cortex (V1)[END_REF].

Consequently, blindsight was eventually classified into two types: type I, the ability to discriminate specific attributes of a stimulus in the contralesional visual field without any awareness of it; and type II: exhibiting objective performance and subjective performance in the blind visual field above chance level, without any conscious detection of the stimuli [START_REF] Brogaard | Type 2 blindsight and the nature of visual experience[END_REF][START_REF] Weiskrantz | Consciousness and commentaries[END_REF].

Diagnosing blindsight implies both accurate measurement of patients' objective performance, and well-controlled and interpretable measurement of their subjective experience, which can depend on the methodology used to assess their awareness [START_REF] Fayel | Characteristics of contralesional and ipsilesional saccades in hemianopic patients[END_REF]. For example, use of a binary scale (i.e. Aware vs. Unaware) may compel a subject to report being "unaware" of a stimulus despite having experienced a minor degree of awareness [START_REF] Overgaard | Visual experience and blindsight: A methodological review[END_REF][START_REF] Overgaard | Seeing without seeing? Degraded conscious vision in a blindsight patient[END_REF]. In one study, the patient G.R. exhibited distinct responses in his blind visual field in function of whether a binary scale (Seen vs. Not Seen) or the Perceptual Awareness Scale (PAS) [START_REF] Ramsøy | Introspection and subliminal perception[END_REF] was used, the latter of which comprises four levels: Clear Experience, Almost Clear Experience, Weak Glimpse and Not Seen. Indeed, when asked to process a stimulus in his contralesional visual field and subsequently report a dichotomous answer (Seen vs. Not Seen), G.R. appeared to exhibit a typical type-I-blindsight profile. However, when assessed by the PAS, he strongly reported some degree of awareness in the presence of stimuli in his contralesional visual field, consistent with a type-II profile. Despite this proven drawback, binary questionnaires are still widely used to study blindsight; indeed, very few researchers use graduated measurements of perception [START_REF] Mazzi | Blind-Sight vs. Degraded-Sight: Different Measures Tell a Different Story[END_REF][START_REF] Overgaard | Seeing without seeing? Degraded conscious vision in a blindsight patient[END_REF]. Thus, our aim in the present study was to assess blindsight in several HH patients using a subjective report on a refined scale to better understand what these patients perceive and experience in their contralesional visual field.

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1.1. Questioning the nature of perceptual experience Among the most controversial issues surrounding blindsight, is that of determining how to understand the nature of patients' perceptual experience in their blind field. Many authors affirm that blindsight is based on unconscious perceptual abilities [START_REF] Leh | Unconscious vision: new insights into the neuronal correlate of blindsight using diffusion tractography[END_REF][START_REF] Sanders | BLINDSIGHT": VISION IN A FIELD DEFECT[END_REF]. However, two other, competing hypotheses have been proposed to explain blindsight (for a review, see [START_REF] Cowey | The blindsight saga[END_REF][START_REF] Hadid | From Cortical Blindness to Conscious Visual Perception: Theories on Neuronal Networks and Visual Training Strategies[END_REF]): that of residual remaining vision; and that of degraded, abnormal vision, which arose from studies on type II blindsight and has been described as a "conscious experience, though of a very different nature to that of normal vision" [START_REF] Kentridge | What is it like to have type-2 blindsight? Drawing inferences from residual function in type-1 blindsight[END_REF][START_REF] Mazzi | Blind-Sight vs. Degraded-Sight: Different Measures Tell a Different Story[END_REF][START_REF] Weiskrantz | Is blindsight just degraded normal vision[END_REF]. Corroborating the residual normal-vision hypothesis, researchers have shown that blindsight could be triggered by the activation of sparse islands of surviving tissue in the primary cortex (V1) [START_REF] Campion | Is blindsight an effect of scattered light, spared cortex, and near-threshold vision[END_REF][START_REF] Fendrich | Speculations on the neural basis of islands of blindsight[END_REF]. However, blindsight aptitudes have also been found in patients that lack a functional primary cortex [START_REF] Ajina | Human blindsight is mediated by an intact geniculo-extrastriate pathway[END_REF][START_REF] Mazzi | Blind-Sight vs. Degraded-Sight: Different Measures Tell a Different Story[END_REF] and in monkeys with complete ablation of V1 [START_REF] Stoerig | Blindsight in man and monkey[END_REF]. In those cases, blindsight seems to be mediated by subcortical pathways that bypass V1, such as the superior colliculus pathway and the dorsal lateral geniculate nucleus (dLGN) pathways.

Based on a meta-analysis of human and monkey data, researchers have concluded that blindsight can indeed occur without any functional portion of the primary visual cortex [START_REF] Stoerig | Blindsight, conscious vision, and the role of primary visual cortex[END_REF]. Moreover, several recent studies have underscored the roles of the ipsilateral hemisphere and the corpus callosum in processing of stimuli in the contralesional visual field of HH patients [START_REF] Bridge | Changes in connectivity after visual cortical brain damage underlie altered visual function[END_REF][START_REF] Celeghin | Intact hemisphere and corpus callosum compensate for visuomotor functions after early visual cortex damage[END_REF][START_REF] Leh | Unconscious vision: new insights into the neuronal correlate of blindsight using diffusion tractography[END_REF].

Currently, there is a lack of consensus on the nature of perceptual experience in blindsight (i.e. totally unconscious vision, residual conscious vision or degraded abnormal vision), which is probably due to the fact that research in this area mainly comprises case studies, often conducted with highly trained patients (e.g. D.B or G.Y; for a review, see [START_REF] Cowey | The blindsight saga[END_REF]) that are familiar with the experimental procedures involved. Therefore, even though blindsight is central to the study of consciousness, there remains a lack of basic epidemiologic data on it (e.g. incidence of blindsight among HH patients).

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Towards population studies of blindsight

Understanding blindsight at the patient-population level would be beneficial both to clinicians working in rehabilitation and to researchers studying the influence of clinical parameters (e.g. the localization, side or size of the brain lesion; or the type of visual-field defect). Only a few studies have been conducted on (small) patient cohorts, and these provide only indirect information on blindsight incidence. For instance, Perenin and Jeannerod found blindsight in six out of eight (75%) patients [START_REF] Perenin | Residual vision in cortically blind hemiphields[END_REF]. Ajina et al. studied the anatomical bases of blindsight in seventeen HH patients, twelve (71%) of whom they classified as "blindsight positive" and five (29%) as "blindsight negative", based on two alternative forced-choice detection tasks related to the contralesional visual field [START_REF] Ajina | Human blindsight is mediated by an intact geniculo-extrastriate pathway[END_REF]. Alternatively, when blindsight was measured in a cohort of twenty patients through a Redundant Signal Effect (RSE) task involving indirect flash presentation to assess visualinformation processing [START_REF] Marzi | Spatial summation across the vertical meridian in hemianopics: A test of blindsight[END_REF], the observed incidence was only 5%. However, in a group study of nineteen patients, in which their pupil responses to visual stimuli presented in their contralesional visual field was used as a proxy for residual visual reflexes, 13 patients (70%) were reported to exhibit blindsight [START_REF] Sahraie | Pupil response as a predictor of blindsight in hemianopia[END_REF]. The above examples indicate that, despite the many articles on blindsight published over the past four decades, there remains a need for systematic investigation of the dissociation between conscious visual detection and non-conscious visual processing abilities in patient groups.

In the work reported here, our aim was two-fold: firstly, to re-evaluate the incidence of blindsight in a population of HH patients by measuring objective performance in two visual tasks; and secondly, to investigate the nature of the perceptual experience in the blind visual field, which has only been examined in single-case studies so far. According to clinical observations, the PAS, a four-level scale based on healthy subjects' spontaneous reports, was not entirely appropriate for evaluating the experience of patients with visual-field defects. Thus, we designed the Sensation Awareness Scale (SAS), including a critical level that does not refer to visual experience. Thus, the complete SAS comprises the following assertions: [1] I did not see anything; [2] I don't think that I saw anything, but I am not sure; [3] I felt something; [4] I saw something; and [5] I clearly saw something and can identify it. Subjective performance was calculated based on the SAS and subsequently compared to the objective performance, which was evaluated through forced-choice detection and identification tasks.

Our experimental design enabled us to reclassify our HH patients according to the dissociation between objective and subjective performance and by different concepts of blindsight described in the literature.

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Thus, we hypothesized that HH (and perhaps other) patients could exhibit the following profiles:

Absence of blindsight: objective performance and subjective performance both at chance level;

Type I blindsight: objective performance above chance level, without any statistically significant subjective sensitivity;

Type II blindsight: objective performance and subjective performance both above chance level, without any conscious detection in the blind visual field; Blindsense: objective performance at chance level combined with a statistically significant subjective performance*; * Note: although this has not yet been defined or labeled as a specific phenomenon, there are literature reports of HH patients that experience some feeling in their blind visual field.

Material and Methods

Participants

We recruited seventeen patients (five females; mean age = 52.8 years, SD = 17.6 years) suffering from left (n = 7) or right (n = 10) HH, induced by a post-stroke unilateral occipital lesion, and seventeen healthy controls (eight females; mean age = 52.2 years, SD = 14.0 years). All 34 participants were right-handed, and all completed a consent form before participation. After neuropsychological and ophthalmologic examinations, only those patients diagnosed with an isolated hemianopia (using Humphrey or Octopus automated perimeters) with corrected-to-normal visual acuity were included in the study. Clinical measurements of visual deficit were collected and analyzed for each patient. Based on a mean deviation obtained for both eyes, a strict threshold (at -30 dB of the mean deviation) was set for the measurements taken at 10° from the fixation point; above this threshold, the visual deficit was considered to be ambiguous (for technical details of the automated measurements, see [START_REF] Sayo | Longitudinal study of visual field changes determined by Humphrey Field Analyzer 10-2 in patients with Retinitis Pigmentosa[END_REF]). Based on this clinical threshold, nine patients (group VIS10; four females; left HH (n = 4); right HH (n = 5)) were excluded from the blindsight performance analyses; thus, these analyses were restricted to the remaining eight patients (group HH; one female; left HH (n = 2); right HH (n = 6))) with no visual detection capacities at 10° of eccentricity. The clinical details and automated visual perimetry values for the patients are presented in Figure 1, Figure 2 andTable 1. p. 9

p. 7 Figure 1: Automated visual perimetry, HH group (n = 8). The axial hash-marks denote increments of 10º (visual) and the greyscale denotes sensitivity (in decibels).

2-column fitting image

Materials

Participants were seated in a dark room and their heads were held in place using a chin piece and a forehead rest at 57 cm from the computer screen (CRT monitor; refresh rate: 85 Hz; resolution: 1280

x 1024 pixels). To control fixation during the experiment, the participants' eye movements were tracked by an EyeTribe.

Stimuli and Procedure

The central fixation point was a red square (1° x 1°). Participants were presented with black letters (X and O; size: 3.5° x 3.5°) at maximal contrast on a gray background, at 10° eccentricity along the horizontal meridian to the left or right of the fixation point. One third of the trials were catch trials with no stimulus presented beside the fixation point. In each session, contralesional and ipsilesional visual fields were stimulated equally. To avoid a habituation bias, the hemifield of stimulus presentation was semi-randomized: thus, a maximum of three successive trials in the same hemifield could be presented. The experiment comprised two sessions: firstly, a detection task; and secondly, a forcedchoice identification task. The two sessions differed only in the type of objective responses; otherwise, the trials were identical on the screen (see Fig. 3). In each session, two types of performance were recorded per trial: objective sensitivity (keyboard answer) and subjective perceptual experience (reported verbally according to the SAS). Objective performance in the detection task was measured on a binary scale as "Something appeared on the screen" (green button) or "Nothing appeared on the screen" (white button). Objective measurements in the identification task were taken by forcedchoice answers: the subjects were asked to specify manually which letter had appeared (X or O button), even if they did not see anything.

Once the objective answer had been recorded, subjective measurements were taken (using the SAS) at the end of a trial. The awareness scales were similar in the two sessions, although they were adapted to the associated objective task. For the identification task, the last two levels of the SAS were modified relative to those described earlier in the introduction:

(4) I saw a letter; and (5) I clearly saw a letter and can identify it. Once the verbal response had been reported by the participant and recorded via keyboard entry by the researcher, the next trial began.

Results

Average group performance

As expected, in the healthy controls (CT) group, the objective and subjective sensitivities were

2-column fitting image

Below, we describe the dissociation between the objective performance and the subjective performance of the HH patients.

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Type I blindsight: Patient P2

Only one of the eight HH patients (P2) could distinguish between the letters X and O in her contralesional visual field, responding at significantly higher than chance level (p < 0.05; Figure 4A dark grey bar). However, P2's objective detection was at chance level, suggesting the presence of type I blindsight. This was confirmed by the subjective experience assessment, which revealed her lack of explicit sensitivity to the presence of stimuli in both tasks (Figure 4B). Indeed, P2 responded at different SAS levels but similarly in the presence or absence of letters on the screen (Figure 5).

Absence of blindsight: Patients P4, P5 and P8

The objective performance and subjective performance of patients P4, P5 and P8 in both tasks did not exceed chance level (Figures 4A &4B). They reported seeing nothing (SAS level 1) during the catch trials or when a letter had been presented in their blind hemifield (Figure 5).

Subjective sensitivity without significant objective identification: Patients P1, P3, P6 and P7

The four remaining HH patients (P1, P3, P6 and P7) did not perform the identification task above chance-level (Figure 4A, dark grey bar); however, they demonstrated significant subjective sensitivity (p < 0.05; Figure 4B). To the best of our knowledge, this profile has never previously been described in the literature. We have named this dissociation between significant subjective performance and chance-level stimulus identification, blindsense. Although P3 demonstrated a significant subjective sensitivity only during the identification task (p = 0.01; light grey bar: P3), P1, P6 and P7 exhibited subjective sensitivity in both tasks. Regarding these patients' SAS responses during the second session, a pattern clearly arose (Figure 5). Their reported awareness when a letter appeared in the blind field is shifted on the scale, mainly at levels 2 (I don't think that I saw anything, but I am not sure) and 3 (I felt something), whereas when nothing appeared on the screen, the patients reported primarily at level 1 (I did not see anything). Interestingly, and as observed in Figure 5, two of the patients (P3 and P7) answered exclusively at level 2 when presented with the stimulus, whereas the other two patients (P1 and P6) answered mainly at level 3 and marginally at level 2. The results for patient P1 differed notably from this group in two regards. Firstly, he was the only patient to both objectively and subjectively detect the presence of stimuli in his blind field (p < 0.05; P1: light grey bar). Secondly, he demonstrated an abnormal reverse performance in identifying the letters, with an objective sensitivity significantly below chance in the identification task (p < 0.05; P1: light grey bar). This atypical result is discussed later as a case of "reverse blindsight". The aforementioned blindsight profiles are summarized in

SAS levels and correctness

We plotted the distributions of correct responses and incorrect responses in the identification task, for each level of the SAS (Appendices, Figure B). Even though P2 successfully identified the letters X and O at a significant level, there was no correlation between a higher reported sensitivity on the scale and the correct answers. In fact, P2 answered mostly at levels 1, 2 and 3, but performed best at level 2.

Concerning the remaining seven patients (P1, P3 to P8), their level of sensitivity was not predictive of their objective performance, either.

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Lesion analysis

As illustrated in Figure 6 Please note that conversely to radiological conventions, here, for convenience, the left hemisphere is presented on the left.

-column fitting image

Discussion

Although the objective tasks that we employed in this study were consistent with the classical format of blindsight experiments, we introduced a novel element in the subjective measure. Indeed, it is not uncommon during clinical examinations that patients with a field defect spontaneously report having a "sensation" in response to a stimulus presented in their blind field-albeit a sensation that they cannot p. 17 qualify as "visual". Despite this common observation, this spontaneously-reported level of awareness has never been integrated in systematic tests. In the present experiment, we developed the SAS, which adds the aforementioned level to the previously reported PAS. Using this procedure, we classified the patients' deficits into different profiles according to the presence or absence of significant sensitivity to the stimulus in the objective measurements and the subjective measurements.

Scarcity of blindsight

Among the eight tested patients, only one patient, P2, met the criteria for blindsight. Indeed, P2 performed the identification task better than chance while being unable to consciously detect the target (detection performance below chance level). Considering that sensitivity on the subjective scale was not significant, P2's profile clearly corresponds to type I blindsight. Using a restrictive threshold to eliminate patients that could residually detect a visual stimulus at 10° of eccentricity, we thus found type I blindsight in only one HH patient. Three patients were unable to identify the letters or to report any subjective experience from the stimulus (Table 2: patients P4, P5 and P8). Since we did not observe any patients with type II blindsight, according to our criteria the HH group showed an incidence of blindsight of 12.5% (one of eight patients) and an absence of blindsight of 37.5% (three of eight patients). Finally, the four remaining patients in the HH group (50%) did not meet the criteria previously described in the literature for either incidence or non-incidence of blindsight. These patients performed at chance-level on the objective task, whereas their sensitivity to the stimulus was significantly above chance when measured on the SAS (see patients P1, P3, P6, and P7 in Table 2).

We have named their profile as blindsense, as all these patients significantly sensed the presence of the stimuli in their blind field without actually identifying the stimuli. We expound on this concept below.

Blindsense

Blindsense patients, exhibiting an unusual dissociation between significant subjective performance and objective performance at chance-level, represented half of our group of complete HH patients. Thus, in our study, a markedly higher number of patients exhibited blindsense than type I blindsight, the latter of which has been extensively discussed in the literature. Blindsense patients do not match the type II blindsight description, since their objective detection and identification performance scores were at chance. However, their SAS answers indicate that their perceptual experiences differed significantly when a letter was presented in their blind field compared to when nothing was presented.

Interestingly, although such a profile has never been described in the literature, it is consistent with a frequently observed clinical reality. Indeed, neuropsychologists specialized in cortical visual impairment have confirmed that many patients report "feeling something" in their contralesional visual field without being able to precisely distinguish or describe the stimuli that they "felt". Here we wish p. 18

to emphasize that, by testing a patient cohort, rather than a single case, and by using an extended awareness scale, we were able to identify the behavioral pattern of blindsense. Assessment of a larger population should enable measurement of the incidence of this phenomenon relative to types I and II blindsight.

One patient, P1, stood out from the blindsense group due to a remarkable result: his identification performance was significantly lower than chance-level, yet he was the only patient to have scored above chance in the binary task, despite being objectively "blind" (based on perimetry). This result suggests that the responses he gave were not completely independent from the stimulus: indeed, P1

pressed the O-key significantly more often when an X appeared on the screen, and vice versa.

However, during the same session, he performed far above chance in the trials involving presentation of a stimulus in the intact visual field. Thus, we can exclude the possibility that his performance was due to an erroneous understanding of the response mapping. One interpretation is that P1 presented a sort of "reverse-blindsight": he processed the visual information, but his behavioral output was somehow reversed. A rather similar phenomenon has been reported in a patient with unilateral neglect [START_REF] Mijović-Prelec | When does "no" really mean "yes"? A case study in unilateral visual neglect[END_REF]. Again, such behavior raises questions on the nature of patients' experiences in their blind visual field, as discussed below.

Awareness measurement

Methods for assessing patients' awareness levels in blindsight experiments remain controversial [START_REF] Overgaard | Visual experience and blindsight: A methodological review[END_REF][START_REF] Weiskrantz | Is blindsight just degraded normal vision[END_REF]. Undeniably, if one claims to demonstrate unconscious abilities, one must provide strong evidence to this effect. In our study, the juxtaposition between the binary scale and the SAS in the same session (detection task) demonstrated that, after patients had answered "nothing appeared on the screen" in a binary forced-choice objective task, they could nonetheless report "not being sure" or "feeling something". Thus, our results highlight two main points: firstly, they confirm that a gradual scale enables more sensitive measurement of the subjective report than does a binary scale; secondly, they suggest that the assessment of detection performance through a binary choice in patients strongly depends on the patients' decision criteria. If researchers choose to use a Yes/No paradigm to assess patient's detection, then one requirement would be to eliminate the response criteria bias through a highly specific psychophysical experimental design, as previously proposed [START_REF] Azzopardi | Is blindsight like normal, near-threshold vision[END_REF]. Moreover, we would like to highlight the fact that the patients in our study, instead answering "I saw something" or "I clearly saw something and can identify it", mainly gave the non-visual answers such as "I am unsure" or "I feel something". This is highly interesting in terms of phenomenological experience. In fact, researchers have traditionally assumed that blindsight is a visual experience, albeit a degraded one. However, as we discuss below, this may not be the case: patients may be aware of a stimulus but unaware of its specific visual nature.

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Nature of the perceptual experience

Is the residual perception experience of patients with visual-field defects triggered by degraded normal vision or is it a "non-visual experience"? [START_REF] Brogaard | Type 2 blindsight and the nature of visual experience[END_REF][START_REF] Kentridge | What is it like to have type-2 blindsight? Drawing inferences from residual function in type-1 blindsight[END_REF][START_REF] Weiskrantz | Is blindsight just degraded normal vision[END_REF]. The profile observed in type II blindsight patients suggests that they do indeed experience a form of degraded normal vision. Corroborating this interpretation, a positive correlation between patients' correct responses in a stimulus-identification task and their level of awareness of the stimulus has previously been reported [START_REF] Overgaard | Seeing without seeing? Degraded conscious vision in a blindsight patient[END_REF]. However, while our blindsense patients displayed above-chance level sensitivity to the stimulus using a subjective scale, they performed at chance level for detection or identification in forced-choice tasks. Thus, it seems that rather than exhibiting normal degraded vision, blindsense patients might instead experience a conscious "non-visual" sensation in response to visual stimulation. These results open a new avenue for future studies on blindsight, suggesting that objective and subjective scales (binary or otherwise) could include non-visual questions to elucidate patients' perceptual experiences in the contralesional visual field. Another critical point to address in the future will be to ascertain any possible differences between left and right HH patients, as our group has previously demonstrated the effect of lesion lateralization on capacities in the contralesional as well as ipsilesional visual field in HH patients [START_REF] Cavézian | Hemisphere-dependent ipsilesional deficits in hemianopia: Sightblindness in the 'intact' visual field[END_REF][START_REF] Chokron | Ipsilesional Deficit of Selective Attention in Left homonymous hemianopia and Left Unilateral spatial Neglect[END_REF]Perez & Chokron, 2014).

Conclusion

Measuring blindsight and perceptual experience in eight patients selected among a group of seventeen post-stroke HH patients, we identified one patient with type I blindsight (12.5%), no cases of type II blindsight (0%), three patients with a total absence of blindsight (37.5%) and most importantly, four patients (50%) exhibiting a never-before described phenomenon, which we have named blindsense.

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Given that blindsight is frequently used to train HH patients [START_REF] Chokron | Ipsilesional Deficit of Selective Attention in Left homonymous hemianopia and Left Unilateral spatial Neglect[END_REF][START_REF] Hadid | From Cortical Blindness to Conscious Visual Perception: Theories on Neuronal Networks and Visual Training Strategies[END_REF]Perez & Chokron, 2014), clinicians and researchers must gain a broader understanding of the subtleties of patients' perception experiences. Further characterization of blindsight types, including the blindsense phenomenon that we have reported here, and understanding of the neuronal correlates of these phenomena, will both require studies of larger cohorts of hemianopic patients.

p. 21 6. Acknowledgments SC is grateful to the Edmond and Benjamin de Rothschild Foundations for their support.

Funding

This research was not funded or financed by any specific grants from public, private or not-for-profit agencies.

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Figure 2 :

 2 Figure 2: Automated visual perimetry, VIS10 group. (n = 9; This group had a detection score higher than -30 dB at 10° in the blind field). The axial hash-marks denote increments of 10º (visual) and the greyscale denotes sensitivity (in decibels).

Figure 3 :

 3 Figure 3: Trial structure of the behavioral task.After the pre-fixation period corresponding to a minimum of 1 second of fixation recorded by the Eyetribe, the fixation square was followed by a gap ranging from 400 ms to 600 ms. A stimulus (letter X or O) was then presented for 200 ms at 10° to the right or left visual field; however, in the catch trials, this time period was substituted with a gap of 200 ms. If the participants did not answer until 600 ms post-stimulus, then a question mark appeared on the screen to remind them that they had to press a key corresponding to their binary answer (detection task: Something appeared vs. Nothing appeared; identification task: X vs. O). Once the subject had given an objective response, they were tested on the awareness scale to obtain the subjective verbal report.2-column fitting image

  significantly above chance (p < 0.05). A Kruskall-Wallis multiple comparison of objective performance among the HH group, VIS10 group (composed of HH patients with spared vision at 10°) and CT group clearly revealed group effects, both in the detection task (H (2, N = 34) = 21.6; p < 0.01) and in the identification task (H (2, N = 34) = 18.6; p < 0.001) (see Appendices, Figure A.). Thus, whereas the HH patients did not significantly differ from the control group in terms of performance in the sighted hemifield Fig A.1, A.2 (red lines), they did perform significantly lower in the blind field than did the controls (Dunn's multiple comparison; p < 0.001 in detection task, p < 0.01 in identification task), Fig A.1, A.2 (blue lines). There was also a group effect regarding subjective performance in both the detection task (H (2, N = 34) = 16.57, p < 0.001) and the identification task (H (2, N = 34 = 17.23; p < 0.001). Dunn's multiple comparison revealed that for both tasks, only the HH patients' mean subjective sensitivity in the blind field was significantly lower than that of the other groups (Fig A.3, A.4; blue lines). Detailed objective and subjective sensitivities of the HH patients for each task are reported in Figure 4. Use of the SAS is represented in Figure 4, which is ordered according to the subjective sensitivity results. p. 12

Figure 4 :Figure 5 :

 45 Figure 4: Comparison of objective sensitivities and subjective sensitivities per patient (HH group, n = 8), per session (detection session: light grey, identification session: dark grey); A. Objective sensitivities (d') for each patient in the detection task (keyboard responses to the binary scale) and the forced-choice task (keyboard responses: X or O) are compared to the chance level (*p < 0.05, performance significantly above chance, with a d' significantly different from a theoretical d' value generated by random responses). B. Subjective-sensitivity AUC values for each patient were calculated by comparing the distribution of the level reported on the SAS between "letter trials" and catch trials. The signed rank test assessed the significance of the difference of the distribution with a p-value of 0.05. (*p < 0.05,

  , we were able to perform neuroanatomic analysis of the lesions in patients P6 and P8. Interestingly, both patients suffered from a complete right homonymous hemianopia. However, P6 exhibited a blindsense profile (i.e. subjective sensitivity without significant objective identification) whereas P8 did not exhibit any sign of blindsight or blindsense. For P6, neuroanatomic analysis revealed a mean lesion volume of 4.98 cm 3 and his lesion involved only Brodmann Areas 17 and 18. In P8, neuroanatomic analysis revealed a mean lesion volume of 14.5 cm 3 and his lesion involved Brodmann Areas 17, 18, 19, 29 and 30.

Figure 6 :

 6 Figure 6: Neuroanatomical lesion analyses: (A) P6, right homonymous hemianopia and blindsense, 3D Flair 8000 sequence acquisition showing the lesion of BA 17 and 18 in the left hemisphere and (B) P8, right homonymous hemianopia with absence of blindsight, 3DTI sequence showing a lesion of BA 17, 18, 19, 29, 30 in the left hemisphere.Please note that conversely to radiological conventions, here, for convenience, the left hemisphere is presented on the left.

  Interestingly, in the present study, a left hemispheric lesion could be associated either with the presence or absence of blindsight, or with blindsense. Accordingly, the association between the lesion lateralization and the subjective experience in the blind visual field is far from being evident and requires further studies among a larger group of HH patients. However, being able to perform a neuroanatomical analysis on patients P6 (blindsense) and P8 (absence of blindsight or blindsense), we found that blindsense in patient P6 was associated to a relatively small lesion that affected only left BA 17 and 18, whereas patient P8, who presented a larger lesion that affected, in addition to left BA 17 and 18, left BA 19, 29 and 30, did not exhibit any sign of blindsense. Our findings suggest that the occurrence of blindsense or blindsight in post-stroke HH patients could require a relatively small lesion that includes, at the very least, loss of the striate cortex, but that spares other occipital areas. Further studies should include analysis of neuroanatomic lesions in larger cohorts of HH patients to elucidate the neuroanatomical correlates of patients' perceptual experience in their contralesional visual field.

Table 1 : Demographic and clinical data

 1

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. M, male; F, female; Onset: the time elapsed between lesion onset and the testing session; L, left; R, right; HH, homonymous hemianopia; "HH group"

Table 2 .

 2

	Patient #	Objective performance	Subjective performance above	Profile
		above chance (p
		<	chance (p < 0.05)
		0.05)
	1	-	✓	Blindsense
	2	✓	-	Type I blindsight
	3	-	✓	Blindsense
	4	-	-	Absence of blindsight
	5	-	-	Absence of blindsight
	6	-	✓	Blindsense
	7	-	✓	Blindsense
	8	-	-	Absence of blindsight

p. 15

TABLE 2 :

 2 Presence and type of blindsight and presence of blindsense, in the HH group, based on the dissociation between objective performance and subjective performance during the

	identification task
	2-column fitting image


domain: math info

text:
Olivier Hudry
email: hudry@telecom-paristech.fr

Antoine Lobstein
email: lobstein@telecom-paristech.fr

Some Results About a Conjecture on Identifying Codes in Complete Suns

Keywords: Graph Theory, Identifying Codes, Locating-Dominating Codes 1

Consider a graph G = (V, E) and, for every vertex v ∈ V , denote by B(v) the set {v} ∪ {u : uv ∈ E}. A subset C ⊆ V is an identifying code if the sets B(v) ∩ C, v ∈ V , are all nonempty and distinct. It is a locating-dominating code if the sets B(v) ∩ C, v ∈ V \ C, are all nonempty and distinct.

Let S n be the graph whose vertex set can be partitioned into two sets U n and V n , where U n = {u 1 , u 2 , . . . , u n } induces a clique, and V n = {v 1,2 , v 2,3 , . . . , v n-1,n , v n,1 } induces an independent set, with edges v i,i+1 u i and v i,i+1 u i+1 , 1 ≤ i ≤ n; computations are carried modulo n. This graph is called a complete sun. We prove the conjecture, stated by Argiroffo, Bianchi and Wagler in 2014, that the smallest identifying code in S n has size equal to n. We also characterize and count all the identifying codes with size n in S n . Finally, we determine the sizes of the smallest locating-dominating codes in S n .

Introduction

This paper was first motivated by a conjecture, stated by Argiroffo, Bianchi and Wagler in [START_REF] Argiroffo | On identifying code polyhedra of families of suns[END_REF], about the size of optimal identifying codes in a special class of graphs, the complete suns. We prove below that this conjecture is true, but we also investigate these graphs further: we characterize and count all their optimal identifying codes, and we determine the size of their optimal locating-dominating codes.

Consider a graph G = (V, E); the order of G is its number of vertices. For every vertex v ∈ V , denote by B(v) the ball of radius one and centre v, i.e., the set {v} ∪ {u : uv ∈ E}. Two vertices u ∈ V , v ∈ V are said to dominate each other if u ∈ B(v), which is equivalent to v ∈ B(u). A vertex z ∈ V is said to separate u and v, or u from v, if it dominates u but not v (note that in this case z = u is possible), or the other way round. A set of vertices Y ⊆ V is said to dominate a vertex u ∈ V (respectively, to separate two vertices u ∈ V and v ∈ V ) if Y contains at least one vertex which dominates u (respectively, which separates u and v).

A subset C ⊆ V is an identifying code if the sets B(v) ∩ C, v ∈ V , are all nonempty and distinct. In other words, every vertex is dominated by C, and every pair of vertices is separated by C. The vertices in C are called codewords and those in V \ C non-codewords. With a slight abuse of language, when the context is clear, we shall often say that "a vertex is dominated", or "a pair of vertices is separated", omitting to add ". . .by the code" or ". . .by such codeword".

When G admits an identifying code, we denote by γ ID (G) the minimum cardinality of an identifying code in G. Any identifying code C with |C| = γ ID (G) is said to be optimal.

Closely related are the locating-dominating codes, which are less demanding: a subset C ⊆ V is a locating-dominating code (LD code for short) if the sets B(v)∩C, v ∈ V \C, are all nonempty and distinct. In other words, every non-codeword is dominated by C, and every pair of non-codewords is separated by C. For LD codes, we shall often say that we separate two vertices also when we put one of them in the code. We denote by γ LD (G) the minimum cardinality of a LD code in G. Any LD code C with |C| = γ LD (G) is said to be optimal.

The motivations for identifying and locating-dominating codes may come, for instance, from fault diagnosis in multiprocessor systems. Identifying codes were introduced in the seminal paper [START_REF] Karpovsky | On a new class of codes for identifying vertices in graphs[END_REF], which establishes general bounds on γ ID , then studies particular classes of graphs, such as hypercubes, trees, or the hexagonal and triangular infinite grids. Locating-dominating codes were introduced in [START_REF] Slater | Domination and location in graphs[END_REF] and further developed in [START_REF] Colbourn | Locating dominating sets in series parallel networks[END_REF], with a focus on complexity and algorithms. Both families of codes now constitute a topic of their own, as shows the bibliography at [START_REF] Lobstein | Watching systems, identifying, locating-dominating and discriminating codes in graphs[END_REF], which contains approximately 340 references. A complete sun S n is a graph whose vertex set can be partitioned into two sets U n and V n , where U n = {u 1 , u 2 , . . . , u n } induces a clique (or complete graph), and V n = {v 1,2 , v 2,3 , . . . , v n-1,n , v n,1 } induces an independent (or stable) set, with edges v i,i+1 u i and v i,i+1 u i+1 , 1 ≤ i ≤ n; computations are carried modulo n. See Figure 1 for n = 4.

u 4 u 1 u 2 v 1,2 v 2,3 v 4,1 v 3,4 V 4 U u 3 4
Complete suns belong to a larger class of graphs, the split graphs, whose vertex set can be partitioned into a clique and an independent set. Complexity results for the identifying code problem in split graphs can be found in [START_REF] Foucaud | The complexity of the identifying code problem in restricted graph classes[END_REF].

In [START_REF] Argiroffo | On identifying code polyhedra of families of suns[END_REF], the authors investigate the minimum cardinality of an identifying code for different families of suns, and in particular they conjecture that for n ≥ 3, γ ID (S n ) = n (note that V n is obvioulsy an identifying code in S n ). In Section 2, we prove that the conjecture is true. We then widen our study of complete suns: in Section 3, we characterize all the optimal identifying codes in S n , and we count them in Section 4; in Section 5, we prove that γ LD (S 3 ) = 3 and γ LD (S n ) = ⌊ 4n+2 5 ⌋ for n ≥ 4. NB: In [1, Fig. 2], the graphs (b) and (c) are erroneously inverted.

Proof of the Conjecture

In the subsequent figures, black circles will represent codewords, white circles will represent non-codewords, and when necessary there will be squares for the vertices with unknown or immaterial status; we do not represent all the edges in the clique, except in Figure 4.

Since V n is an identifying code, all we have to prove is that γ ID (S n ) ≥ n. Let C n be an identifying code in S n , optimal or not. When we speak of consecutive vertices in U n or V n , "consecutive" refers to the subscripts. Proof. (a) If v i,i+1 is a single, then u i and u i+1 are not separated by the code, since v i-1,i and v i+1,i+2 are the only vertices separating u i and u i+1 . (b) If v i-1,i , v i,i+1 and v j-1,j , v j,j+1 form a quartet, then u i and u j are not separated, with j ∈ {1, . . . , n} \ {i -1, i, i + 1}. ♦

A vertex v i,i+1 such that v i-1,i / ∈ C n and v i+1,i+2 / ∈ C n is called a single. A sequence of two consecutive non-codewords v i-1,i , v i,i+1
V n . u 2 v 2,3 v 1,2 u 2 v 2,3 U n V n v 1,2 U n ... ... V n ...
Proposition 2 For all integers n ≥ 3, we have:

γ ID (S n ) ≥ n.
Proof. Using Lemma 1, we study the possible sequences of codewords and non-codewords in V n . In particular, no sequence of three consecutive noncodewords exists in V n .

There is at least one codeword in V n , because the vertices in U n are separated, and without loss of generality (wlog), we assume that v 1,2 ∈ C n . Then by Lemma 1(a) v n,1 ∈ C n or v 2,3 ∈ C n : wlog, we take v 2,3 ∈ C n . We can then alternate the codewords and the non-codewords in V n as in Figure 2; in Figure 2(a), we have:

• Sequences of consecutive codewords of various lengths, with at least one sequence, and the lengths at least two by Lemma 1(a).

• One duet; by Lemma 1(b), there is at most one duet.

• Sequences of isolated non-codewords; their number can range between zero and ⌊ n-4 3 ⌋. In Figure 2(b), we have only isolated non-codewords in V n ; their number can range between zero and ⌊ n 3 ⌋ (but as we shall see later on, exactly one [isolated] non-codeword in V n is impossible).

We denote by q the number of codewords in V n ; then there are nq non-codewords in V n , distributed according to Figure 2(a) or (b). These non-codewords must be dominated and separated by codewords in U n . It is straightforward to observe that each isolated non-codeword requires (at least) one codeword in U n (in order to be dominated), that the duet requires (at least) two codewords in U n (so that its two vertices are dominated and separated), and that these tasks do not overlap. So the nq non-codewords in V n require (at least) nq codewords in U n , and the total number of codewords is (at least) n. ♦ Thus, the conjecture is proved:

U n V n u i u i+2 v i+2,i+3 V n U n u i u i+2 v i+2,i+3 v i-1,i v i-1,i ... ... ...
Corollary 3 For all integers n ≥ 3, we have: γ ID (S n ) = n. ♦

Characterization of the Optimal Codes

We construct all the optimal identifying codes in S n by first choosing the configuration of codewords/non-codewords in V n ; this configuration is one of the two given in Figure 2. Then we complete the code by choosing the remaining codewords in U n : for each isolated non-codeword in V n , we arbitrarily choose one of the two vertices in U n that dominates it, see Figure 3(a); for each duet in V n , we arbitrarily choose two vertices among the three vertices in U n that dominate at least one vertex of the duet, see Figure 3(b)-(c). We claim that, except in some cases that will be described below, the code C n thus constructed is identifying. Since it contains n elements, it is optimal.

(1) Obviously, all the vertices are dominated by C n .

(2) The vertices in U n are separated between themselves: two consecutive vertices are separated because there is no single in C n ; two non-consecutive vertices are separated because there is no quartet in C n .

(3) The vertices in U n are separated from the vertices in V n : indeed, unless (a) there is exactly one codeword u i in U n , or (b) there are exactly two codewords u i , u i+1 in U n , the codewords in U n perform this separating task. In Case (a) however, u i and the codeword in V n that u i dominates are not separated (cf. Figure 3(a)); this case can be avoided only if we do not take exactly n -1 codewords in V n . In Case (b), if u i and u i+1 have been put in the code to dominate and separate v i,i+1 and v i+1,i+2 , then u i+1 and v i,i+1 are not separated, (cf. Figure 3(b)), but by taking rather u i and u i+2 , like in Figure 3(c), we can overcome this difficulty. (4) The vertices in V n are separated between themselves: (a) The codewords in V n are separated from all the other vertices in V n by themselves.

(b) The non-codewords in V n are separated between themselves by the codewords in U n that dominate them. Summarizing: in order to construct all the optimal identifying codes in S n , we choose a configuration of codewords and non-codewords in V n such as given by Figure 2, with the exception of the configuration with exactly n -1 codewords in V n . We complete the code with vertices in U n , in the way described in the first paragraph of this Section, being careful, in the case when v i,i+1 and v i+1,i+2 are the only non-codewords in V n , to take u i and u i+2 as codewords, not u i and u i+1 nor u i+1 and u i+2 .

Finally, Figure 4 gives all the nonisomorphic optimal identifying codes in S n , for n = 3, n = 4, n = 5 and n = 6, with decreasing number of codewords in V n . Figure 5 gives, for n = 7, the four possible configurations for V n , giving respectively one, one, three and one nonisomorphic optimal identifying codes in S [START_REF] Lobstein | Watching systems, identifying, locating-dominating and discriminating codes in graphs[END_REF] .

An open question is to determine the number of nonisomorphic optimal identifying codes in S n for n > 7, but in the following Section, we manage to count all the optimal identifying codes in S n , isomorphic or not. [START_REF] Foucaud | The complexity of the identifying code problem in restricted graph classes[END_REF] Counting the Optimal Codes 4.1 The case when there is no duet in V n Consider the set V n = {v 1,2 , v 2,3 , . . . , v n-1,n , v n,1 } and an optimal identifying code C n such that there is no duet in V n ; for i ∈ {1, . . . , n}, the set

1,2 v n,1 v H k B 1 B 2 B k H 1 H 0 0 = H 0 B 2 B 1 0 = H 1 (c) V n ... ... duet n,1 v 1,2 v B 1 H 0 H 1 H k (a) V n ... V n (b) ...
{v i-1,i , v i,i+1 , v i+1,i+2 } is called a brick if v i-1,i ∈ C n , v i,i+1 /
∈ C n , and v i+1,i+2 ∈ C n . Let k be the number of bricks is V n ; k is also the number of (isolated) non-codewords in V n , and we have seen previously that k ∈ K n = {0} ∪ {2, 3, . . . , ⌊ n 3 ⌋}. Two sets of vertices of V n are said to be consecutive if both consist of consecutive vertices in V n , and if the last vertex of one set and the first vertex of the other set are consecutive.

Assume first that v 1,2 does not belong to any brick; in particular, v 1,2 ∈ C n . Going from v 2,3 to v n,1 , we can partition V n \ {v 1,2 } into 2k + 1 consecutive sets in the following way (see Figure 6(a)):

V n \ {v 1,2 } = H 0 ∪ B 1 ∪ H 1 ∪ . . . ∪ H k-1 ∪ B k ∪ H k , (1)
where the sets H i contain only codewords and can have size zero, and the sets B i are bricks (of size 3). Let

h i = |H i |.
The partitions of type (1) are characterized by the sizes of the sets H i , and their number is given by the number of solutions of

k i=0 h i = n -1 -3k, 0 ≤ h i , (2)
which is known to be n-2k-1 k , see, e.g., [START_REF] Šek | Invitation to Discrete Mathematics[END_REF]. We have seen that for each isolated non-codeword v j,j+1 in V n , we can choose as codeword in U n either u j or u j+1 . So, under these conditions (no duet, no brick containing v 1,2 ), the number of optimal identifying codes is

α n = k∈K n-1 2 k n -2k -1 k = ⌊ n-1 3 ⌋ k=0 2 k n -2k -1 k -2(n -3). ( 3
)
Assume next that v 1,2 belongs to one brick (hence k = 0), and let k * = k -1.

There are three possibilities according to the position of v 1,2 in the brick (leftmost, middle or rightmost). Then we can apply the previous argument, with the following differences nonetheless: k * ∈ {1, 2, . . . , ⌊ n 3 ⌋ -1}, and the partition is done on the n -3 vertices surrounding the brick containing v 1,2 , with k * bricks and k * + 1 sets H i ; see Figure 6(b). So (2) becomes

k * i=0 h i = n -3 -3k * , 0 ≤ h i , which admits n-2k * -3 k *
solutions. Keeping in mind that we have three positions for v 1,2 inside its brick, and two possibilities in the choice of the codeword in U n which dominates v 1,2 , we can see that the number of optimal identifying codes is in this case is given by

β n = 3 × 2 × ⌊ n-3 3 ⌋ k * =1 2 k * n -2k * -3 k * . ( 4
)
So, in the absence of duet, the number of optimal identifying codes in S n is equal to

δ n = α n + β n , (5)
where α n and β n are given by ( 3) and (4), respectively.

The case when there is one duet in V n

Now we assume that C n is an optimal identifying code such that there is one duet in V n (we have seen previously that there can be at most one duet). First we choose the position of the duet; we have n possibilities. The duet, together with its left and right neighbouring codewords, occupies four vertices. Then we proceed as before, with k bricks, k ∈ {0, 1, 2, . . . , ⌊ n- 4 3 ⌋} (no restriction on the number of bricks when there is a duet): we have to choose a partition on the n -4 vertices surrounding the duet and its left and right neighbours (see Figure 6(c)), so we have to consider the number of solutions of

k i=0 h i = n -4 -3k, 0 ≤ h i , which is n-2k-4 k
. We have seen that there is no choice for the two codewords in U n taking care of the duet, and we can conclude that the number of optimal identifying codes is given by

ω n = n × ⌊ n-4 3 ⌋ k=0 2 k n -2k -4 k (6)
in case of a duet.

Conclusion

Gathering (3), ( 4), ( 5) and ( 6) gives the total number of optimal identifying codes, which is

Φ n = ⌊ n-1 3 ⌋ k=0 2 k n -2k -1 k + 6 • ⌊ n-3 3 ⌋ k=1 2 k n -2k -3 k + n • ⌊ n-4 3 ⌋ k=0 2 k n -2k -4 k -2(n -3). ( 7
)
The first values given by ( 7) are Φ 3 = 1, Φ 4 = 5, Φ 5 = 6, Φ 6 = 19, Φ 7 = 50, Φ 8 = 89 and Φ 9 = 160.

In order to obtain an asymptotic estimation of Φ n , we need to approximate M n =max k∈{0,...,⌊n/3⌋} 2 k n-2k k . Setting N = n-2k and ε = k/N leads to study max ε∈]0,1[ 2 N ε N N ε , and, using a standard approximation on the binomial coefficient (see, e.g., (2.4.3) in [START_REF] Cohen | Covering Codes[END_REF]), to study max ε∈]0,1[ N (ε + H(ε)), where H(x) = -x log 2 (x) -(1x) log 2 (1x) is the (binary) entropy function. Going back to n, we have to search for max ε∈]0,1[ n ε+H (ε) 1+2ε , that is, to search for max ε∈]0, 1[ ε+H(ε) 1+2ε . This maximum is reached for ε 0 ≈ 0.41025 and k 0 ≈ 0.22535n, and is approximately 0.76181, which leads finally to M n 2 0.76181n . Since n * = 2n is the order of the complete sun S n , we have, in terms of order, about 2 0.38091n * different optimal identifying codes in the complete sun.

Let us now consider briefly the problem of finding graphs of order n * with a large number of optimal identifying codes, see [START_REF] Honkala | On the number of optimal identifying codes in a twin-free graph[END_REF]. The complete suns are graphs for which the optimal identifying codes have size equal to half their order, and as such could be candidates, since n * γ ID (G) , the number of all subsets of vertices of size γ ID (G) in any graph G of order n * , is maximum when γ ID (G) = n * /2. However, the above result, 2 0.38091n * , is modest when compared to the best known result, namely 2 0.77003n * ([5, Th. 11]). a triangle is empty if it contains no codeword. Now, with this terminology, Lemma 4(b) states that there is at most one empty triangle ▽, and Lemma 4(c) states that there is no empty triangle △.

v i+4,i+5 v i+1,i+2 V n U n u i+5 u i+2 u i+1 v i,i+1
Amidst all the LD codes in S n with size γ LD (S n ), we take one, and we denote it by C n . Assume that V n ∩ C n = ∅; then the vertices in U n cannot be separated, unless all of them but one are codewords. This implies

|C n | ≥ n -1 ≥ ⌊ 4n+2
5 ⌋, for n ≥ 5. So from now on, we can take for granted that V n contains at least one codeword.

Step 1. Any set {v i,i+1 , v i+1,i+2 , . . . , v i+k-1,i+k , u i+1 , u i+2 , . . . , u i+k }, i ∈ {1, . . . , n}, k ∈ {1, . . . , n}, is denoted by T i,2k and is called a 2k-slice. In this first step, our goal is to show that for all i ∈ {1, . . . , n},

|T i,10 ∩ C n | ≥ 4, (9)
with the exception of three cases which will be specifically treated. We are going to study how many codewords can be in V n ∩ T i,10 , for any i.

(1) If there are four or five codewords in V n ∩ T i,10 , then (9) is satisfied.

(2) Next, assume that there are three codewords in V n ∩ T i,10 ; then, because there are four triangles △ in T i,10 , at least one more codeword is necessary in U n ∩ T i,10 , and ( 9) is satisfied.

(3) We now turn to the case when there are two codewords (among five vertices) in V n ∩ T i,10 ; all the 5 2 = 10 configurations require at least two more codewords in U n ∩ T i,10 , and again (9) is satisfied.

(4) If |V n ∩ C n ∩ T i,10 | = 1,
then, among the five possibilities, one can see, in particular using Lemma 4, that only one is possible with only two codewords in U n ∩ T i,10 , see Figure 8, where one can see an empty triangle ▽.

(5) If V n ∩ C n ∩ T i,10 = ∅, then, among the 5 3 = 10 configurations for U n ∩ T i,10 , one can see, again using Lemma 4, that only two of them can possibly lead to LD codes with three codewords in U n ; they are given in Figure 9. Note that both contain an empty triangle ▽.

So far, the above five cases show that for all i ∈ {1, . . . , n}, the set C n ∩ T i,10 has size either (a) at least 4, or (b) equal to 3; in the latter case, only three configurations are possible, given by Figures 8 and9, and at most

V n U n v i,i+1 v i+4,i+5 u i+1 v i+1,i+2 V n u i+2 v i+4,i+5 u i+4 U n v i+1,i+2 u i+2 u i+5 u i+5 u i+1 v i,i+1 (a) (b)
Figure 9: The 10-slice T i,10 with three codewords, all of them in U n .

V n U n

Figure 10: Two 10-slices containing each three codewords. one of them can appear, at most once, because they all contain an empty triangle ▽; more precisely, this is true only if no intersection is allowed: see for instance Figure 10, where the configurations of Figures 8 and9(a) coexist, but share eight vertices. Since our goal is to split S n into disjoint 10-slices, we shall go on saying, with a slight abuse of language, that at most one of them appears at most once.

Moreover, in Figure 8, u i ∈ C n , by Lemma 4(c), and v i+5,i+6 ∈ C n , in order to avoid the pattern of Lemma 4(a). The same is true in Figure 9(a), by Lemma 4(d)-(b), as well as in Figure 9(b), by Lemma 4(c)-(b). We shall say that in these three cases, the left outside and right outside neighbours of T i,10 must be codewords.

We can now proceed to Step 2, in which we show how, according to the congruences of n modulo 5, we can split S n into 10-slices and one smaller slice, and we prove the lower bound.

Step 2. Given C n , for k ∈ {1, . . . , n} and j ∈ {0, . . . , 2k}, we say that a 2k-slice containing exactly (respectively, at least) j codewords is a (2k, j)slice (respectively, a (2k, ≥ j)-slice). With this notation, what we have just proved in Step 1 is that our graph, when arbitrarily split into disjoint 10-slices (plus one r-slice, r ∈ {0, 2, 4, 6, 8}), can only contain at most one (10, 3)-slice, and (10, ≥ 4)-slices. Moreover, any (10, 3)-slice has its left and right outside neighbours in the code, which in particular helps to show that

V n U n u i+5 V n U n v i,i+1 v i+2,i+3 V n v i+4,i+5 v i+3,i+4 v i,i+1 u i+1 v i+1,i+2 v i+4,i+5 v i+3,i+4 v i+2,i+3 v i+1,i+2 v i+4,i+5 v i+3,i+4 u i+1 U n u i+5 u i+1 v i+2,i+3 v i+1,i+2 u i+2 v i,i+1 u i+2 u i+4 u i+5 (a) (b) (c)
The following lemma gives information on the (10, 4)-slices when there is a (10, 3)-slice in S n .

Lemma 5 Assume that we have one (10, 3)-slice T j,10 in S n . Let T i,10 be any (10, 4)-slice with no intersection with T j,10 . Then:

(a) It is impossible to have v i,i+1 ∈ C n and u i+5 ∈ C n simultaneously; (b) If v i,i+1 ∈ C n , then v i+5,i+6 ∈ C n ; (c) If u i+5 ∈ C n , then u i ∈ C n .
Proof. The fact that we have a (10, 3)-slice forbids that an empty triangle ▽ appears in T i,10 . So in Figure 11, there is at least one codeword in each of the triangles ▽. Also, by Lemma 4(c), there is at least one codeword in the triangles △. We represent only the triangles which are used in the proof. (a) Assume that v i,i+1 ∈ C n and u i+5 ∈ C n , see Figure 11(a). If v i+1,i+2 ∈ C n or u i+1 ∈ C n , then v i+2,i+3 and v i+3,i+4 are still to be dominated and separated, which is impossible with only one additional codeword.

So v i+1,i+2 / ∈ C n , u i+1 / ∈ C n , which implies that u i+2 ∈ C n (triangle △).
Then the remaining codeword is at the intersection of the two triangles ▽, v i+3,i+4 , but we have the forbidden pattern of Lemma 4(a).

(b) Assume that v i,i+1 ∈ C n and v i+5,i+6 / ∈ C n , see Figure 11(b). By (a) just above, we have u i+5 / ∈ C n . Now necessarily v i+4,i+5 ∈ C n (triangle ▽). Then by Lemma 4(a), u i+4 ∈ C n or v i+3,i+4 ∈ C n , and there remains only one codeword, which cannot dominate and separate v i+1,i+2 and v i+2,i+3 .

(c) Assume that u i+5 ∈ C n and u i / ∈ C n , see Figure 11(c). By (a), we

have v i,i+1 / ∈ C n , which implies u i+1 ∈ C n (leftmost triangle △). But v i,i+1 and v i+1,i+2 are not separated yet, so v i+1,i+2 ∈ C n or u i+2 ∈ C n . If it is v i+1,i+2
, there remain two empty disjoint triangles, one △ and one ▽, with only one more codeword available. If it is u i+2 , then the fourth codeword is the vertex at the intersection of the two triangles ▽, v i+3,i+4 , but we have the forbidden pattern of Lemma 4(a). ♦

Remark 6

The previous lemma is still true if we weaken its condition, by assuming now that there is in S n an empty triangle ▽ with no intersection with T i,10 .

We are now going to consider the different congruences of n modulo 5.

Proposition 7 For every integer k ≥ 1, we have γ LD (S 5k ) = 4k.

Proof. The upper bound (8) gives γ LD (S 5k ) ≤ 4k. By (10), we can assume that k > 1. The only way to have fewer than 4k codewords in a LD code is to have one (10, 3)-slice and k -1 (10, 4)-slices, in which case we have 4k -1 codewords. Assume wlog that the (10, 3)-slice is T 1,10 , and consider the k -1 (10, 4)-slices T 6,10 , T 11,10 , . . ., T 5k-4,10 . We have observed just before inequality (10) that T 1,10 has its right outside neighbour in the code, i.e., v 6,7 ∈ C n . Then, by Lemma 5(b), v 11,12 ∈ C n , . . ., v 5k-4,5k-3 = v n-4,n-3 ∈ C n , which implies, by Lemma 5(a) applied with i = n -4, that u 1 / ∈ C n , contradicting the fact that T 1,10 has its left outside neighbour in the code. Note that we could have gone the other way round, using Lemma 5(c): T 1,10 has its left outside neighbour in the code, so u 1 ∈ C n , u 5k-4 ∈ C n , . . ., u 11 ∈ C n , which implies that v 6,7 / ∈ C n , which contradicts the fact that T 1,10 has its right outside neighbour in the code.

Anyway, we have proved that the lower and upper bounds coincide. ♦ Proposition 8 For every integer k ≥ 1, we have γ LD (S 5k+1 ) = 4k + 1.

Proof. The upper bound [START_REF] Šek | Invitation to Discrete Mathematics[END_REF] gives γ LD (S 5k+1 ) ≤ 4k + 1. Assume first that k = 1. Is |C 6 | = 4 possible? If there is a (10, 3)-slice in S 6 , then, since its left and right outside neighbours are codewords, we have five codewords. So we can exclude a (10, 3)-slice.

Consider a (2, ≥ 1)-slice (there must be one!). Wlog, it is T 1,2 . Then the 10-slice T 2,10 is a (10, ≥ 4)-slice, and we have at least five codewords. So γ LD (S 6 ) = 5.

From now on, k > 1. Assume first that there is a (10, 3)-slice, say T 2,10 , and consider the 2-slice T 1,2 . Because u 2 , the left outside neighbour of T 2,10 , is a codeword, T 1,2 is a (2, ≥ 1)-slice, and because there can be only one (10, 3)-slice, the k -1 10-slices T 7,10 , . . ., T 5k-3,10 are (10, ≥ 4)-slices. This implies, if we want to get at most 4k codewords, that T 1,2 is a (2, 1)-slice, that the k -1 10-slices other than T 2,10 are (10, 4)-slices, and that we have exactly 4k codewords. Because the right outside neighbour of T 2,10 , v 7,8 , is a codeword, and using repeatedly Lemma 5(b), we have v 7,8 ∈ C n , v 12,13 ∈ C n , . . ., v 5k+2,5k+3 = v 1,2 ∈ C n , i.e., T 1,2 has its two elements, u 2 and v 1,2 , in C n , a contradiction. So there is no (10, 3)-slice in S 5k+1 .

As in the case k = 1, consider a (2, ≥ 1)-slice, say T 1,2 . Then the k 10-slices T 2,10 , T 7,10 , . . ., T 5k-3,10 , which are (10, ≥ 4)-slices, lead to at least 4k + 1 codewords.

So no LD code with 4k elements exists in S 5k+1 . ♦ Proposition 9 For every integer k ≥ 1, we have γ LD (S 5k+2 ) = 4k + 2.

Proof. The upper bound [START_REF] Šek | Invitation to Discrete Mathematics[END_REF] gives γ LD (S 5k+2 ) ≤ 4k + 2. Assume first that k = 1. Is |C 7 | = 5 possible? If there is a (10, 3)-slice in S 7 , say T 1,10 , then, since its two outside neighbours, u 1 and v 6,7 , are codewords, we have already five codewords. So u 7 / ∈ C 7 and v 7,1 / ∈ C 7 . Then, taking the configurations of the (10, 3)-slice as given by Figures 8 or 9(b), where neither v i,i+1 (here, v 1,2 ) nor u i+1 (here, u 2 ) are codewords, we can see that v 7,1 and v 1,2 are not separated, i.e., at least six codewords are necessary. If we use the configuration of Figure 9(a), where v i+4,i+5 and u i+5 (here, v 5,6 and u 6 ) are not codewords, we have the pattern forbidden by Lemma 4(a). So no (10, 3)-slice is possible. If we have a (10, 4)-slice, see the general case with k (10, 4)-slices. If we have a (10, 5)-slice, and since the 4-slice must be a (4, ≥ 1)-slice (existence of a △), we have also at least six codewords. So γ LD (S 7 ) = 6.

From now on, k > 1. Can we have 4k + 1, or fewer, codewords? Assume first that there is a (10, 3)-slice, say T 1,10 . Its outside neighbours, u 1 and v 6,7 , are codewords.

Consider the 4-slice T 6,4 . Because v 6,7 ∈ C n , and because there is a triangle △ in T 6,4 , T 6,4 is a (4, ≥ 2)-slice. The k -1 10-slices T 8,10 , . . ., T 5k-2,10 are (10, ≥ 4)-slices, so that necessarily we have: one (10, 3)-slice, one (4, 2)-slice, k -1 (10, 4)-slices, and exactly 4k + 1 codewords.

Assume first that T 1,10 is given by Figure 9(a), so that v 5,6 / ∈ C n , u 6 / ∈ C n . The fact that u 1 ∈ C n ∩ T 5k-2,10 implies, by Lemma 5(c), that u 5k-2 ∈ C n , . . ., u 8 ∈ C n , so that the two non-codewords in T 6,4 must be u 7 and v 7,8 . Now the non-codewords v 5,6 , u 6 , u 7 and v 7,8 form a forbidden pattern.

Assume next that we are in one of the configurations of Figures 8 or 9(b): in particular, v 1,2 / ∈ C n and u 2 / ∈ C n . Now we consider the 4-slice T n-1,4 , which contains the codeword u 1 , and a second codeword because it contains a triangle ▽, and there is already an empty ▽ in T 1,10 . Since v 6,7 ∈ C n ∩T 6,10 , we have, as previously seen several times, v 11,12 ∈ C n , . . ., v 5k+1,5k+2 = v n-1,n ∈ C n , i.e., C n ∩T n-1,4 = {u 1 , v 5k+1,5k+2 }. But now the non-codewords u 5k+2 , v 5k+2,1 , v 1,2 and u 2 form a forbidden pattern.

Both cases lead to a contradiction, showing that there is no (10, 3)-slice in S 5k+2 .

Then, to have 4k + 1 codewords, we can only have k (10, 4)-slices and one (4, 1)-slice. Take one codeword v ℓ,ℓ+1 in V n (we know that there is one), and consider the 4-slice T ℓ,4 . Because there is a triangle △ in T ℓ,4 , there are at least two codewords in T ℓ,4 and we cannot reach 4k + 1 codewords. ♦ Proposition 10 For every integer k ≥ 1, we have γ LD (S 5k+3 ) = 4k + 2.

Proof. The upper bound [START_REF] Šek | Invitation to Discrete Mathematics[END_REF] gives γ LD (S 5k+3 ) ≤ 4k + 2. Observe that any 6-slice is necessarily a (6, ≥ 2)-slice: if there is only one codeword, it must be the middle vertex in U n , because it is at the intersection of the two triangles △, but then there is the forbidden pattern of Lemma 4(d).

Assume first that k = 1. Is |C 8 | = 5 possible? If there is a (10, ≥ 4)-slice, then there are at least 4 + 2 = 6 codewords. So we assume that there is a (10, 3)-slice. Its two outside neighbours are codewords, but there is still one empty triangle △ in the 6-slice (and also one ▽), so at least one additional codeword is required. We can conclude that γ LD (S 8 ) = 6.

Assume next that k > 1. The only way to have at most 4k +1 codewords is to have one (10, 3)-slice, k-1 (10, 4)-slices and one (6, 2)-slice, and we have exactly 4k+1 codewords. Wlog, the (10, 3)-slice is T 4,10 , and we consider the (6, 2)-slice T 1,6 . Then u 4 , the left outside neighbour of T 4,10 , is a codeword; the second codeword in T 1,6 must belong to the triangle △ consisting of u 2 , v 2,3 , u 3 , but each of the three possibilities leads to a forbidden pattern, either by Lemma 4(a) or (b) (since an empty triangle ▽ already appears in T 4,10 ). This concludes the case k > 1.

♦ Proposition 11 For every integer k ≥ 1, we have γ LD (S 5k+4 ) = 4k + 3.

Proof. The upper bound [START_REF] Šek | Invitation to Discrete Mathematics[END_REF] gives γ LD (S 5k+4 ) ≤ 4k + 3. If we consider any 8-slice, say T 1,8 , it is straigthforward to see (for instance by checking the nine possibilities given by the two disjoint triangles △) that the only possibility with two codewords in T 1,8 that avoids any forbidden pattern is C n ∩ T 1,8 = {u 3 , u 4 }, which however contains an empty triangle ▽, incompatible with a (10, 3)-slice.

If k = 1, to have only six codewords would lead to a (10, 3)-slice with a (8, 3)-slice, or a (10, 4)-slice with a (8, 2)-slice. In the first case, if the (10, 3)slice is T 5,10 , its outside neighbours u 5 and v 1,2 are codewords, the third codeword in T 1,8 is u 3 , because it is at the intersection of two triangles △ with no codeword yet, but then the forbidden pattern of Lemma 4(d) appears. In the second case, if the (10, 4)-slice is T 5,10 , and, remembering that the only two codewords in T 1,8 are u 3 and u 4 , we can see immediately that u 1 ∈ C 9 (to avoid an empty triangle △) and v 5,6 ∈ C 9 (to avoid a second empty triangle ▽). This however contradicts, for n = 9 and i = 5, Lemma 5(a), used together with Remark 6, since there is an empty triangle ▽ outside T 5,10 . So γ LD (S 9 ) = 7. Now k > 1, and we assume that there is a (10, 3)-slice, say T 5,10 . Then, to obtain 4k + 2 codewords, we have one (8, 3)-slice and k -1 (10, 4)-slices.

Figure 1 :

 1 Figure 1: The complete sun S 4 .

Figure 2 :

 2 Figure 2: Codewords and non-codewords in V n .

Figure 3 :

 3 Figure 3: (a) How to treat isolated non-codewords in V n ; (b) and (c) How to treat a duet.

Figure 4 : 7 V 7 Figure 5 :

 4775 Figure 4: All the nonisomorphic optimal identifying codes. V 7 V 7

Figure 6 :

 6 Figure 6: Examples of partitions of V n : (a) no duet, v 1,2 not in a brick; (b) no duet, v 1,2 in the brick B; (c) one duet.

Figure 8 :

 8 Figure 8: The 10-slice T i,10 with three codewords, one in V n and two in U n .

Figure 11 :

 11 Figure 11: Illustration of Lemma 5, the (10, 4)-slice T i,10 .

  is called a duet; two duets with no vertices in common form a quartet.

	Lemma 1 Let C n be an identifying code in S n . Then:
	(a) There is no single in V n ;
	(b) There is at most one duet in V n ; equivalently, there is no quartet
	in

The first two values of n are easy to study, and we have: γ LD (S 3 ) = 3, γ LD (S 4 ) = 3. From now on, we assume that n ≥ 5.

We first give five constructions, according to the congruences of n modulo 5, proving that for n ≥ 5, we have:

These constructions are given by Figure 7 and are straightforward to check (remember that only the non-codewords have to be tested).

We next turn to the lower bound which we want to prove:

The following lemma, where (a) and (b) match Lemma 1 for identifying codes, will prove useful.

Lemma 4 Let C n be a LD code in S n , and i ∈ {1, . . . , n}. Then:

Proof. (a) Because u i and u i+1 would not be separated. (b) Assume on the contrary that there exists j ∈ {1, . . . , n}, j = i, such that C n ∩ {v j-1,j , u j , v j,j+1 } = ∅; then u i and u j are not separated.

(c) Because v i,i+1 must be dominated. (d) Because v i,i+1 and v i+1,i+2 must be separated. ♦

In the following, we shall say that the sets {v i-1,i , u i , v i,i+1 } and {u i , v i,i+1 , u i+1 } are triangles, denoted by ▽ and △, respectively. We shall say that Consider the 8-slice T 1,8 . Then the right outside neighbour of T 5,10 , v 10,11 ∈ T 10,10 , is a codeword, and so are, by Lemma 5(b), v 15,16 , . . ., v 5k+5,5k+6 = v 1,2 . Also, the left outside neighbour of T 5,10 , u 5 , is a codeword. So we have {v 1,2 , u 5 } ⊂ C n ∩ T 1,8 , and, exactly as above, in the first case for k = 1, we are led to a contradiction. Conclusion: there is no (10, 3)-slice in S 5k+4 . Now the only way to have at most 4k + 2 codewords is to have k (10, 4)slices and one (8, 2)-slice.

Consider the (8, 2)-slice T 1,8 ; we have, as in the second case for k = 1, u 1 ∈ C n and v 5,6 ∈ C n . Then again, because we are allowed to use Lemma 5 thanks to the existence of an empty triangle ▽ in the (8, 2)-slice, we have, by Lemma 5(b), v 10,11 ∈ C n , . . ., v 5k,5k+1 = v n-4,n-3 ∈ C n , but the fact that v n-4,n-3 ∈ C n , together with u 1 ∈ C n , leads to a contradiction by Lemma 5(a), applied to i = n -4.

♦

We gather all our results in the following proposition.

Proposition 12 (a) γ LD (S 3 ) = 3; (b) For every integer n ≥ 4, γ LD (S n ) = 4n+2

5

.

♦

It would be interesting to do the same as for identifying codes and be able to characterize all the optimal locating-dominating codes in S n , but from what we have seen above, it seems quite intricate to try to do so, and this remains an open problem.