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Construction of Codes Identifying Sets of Vertices

Sylvain Gravier, Julien Moncel
<span title="2005-03-08">2005</span> <i title="The Electronic Journal of Combinatorics"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/v5dyak6ulffqfara7hmuchh24a" style="color: black;">Electronic Journal of Combinatorics</a> </i> &nbsp;
Here we construct graphs on $n$ vertices having a $(1,\le \ell)$-identifying code of cardinality $O\left(\ell^4 \log n\right)$ for all $\ell \ge 2$.  ...  In this paper the problem of constructing graphs having a $(1,\le \ell)$-identifying code of small cardinality is addressed.  ...  )-identifying code (Lemma 4).  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.37236/1910">doi:10.37236/1910</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/g4dbfoa2ivgjjadmpod5o2niki">fatcat:g4dbfoa2ivgjjadmpod5o2niki</a> </span>
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On graphs on n vertices having an identifying code of cardinality ⌈log2(n+1)⌉

Julien Moncel
<span title="">2006</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/lx7dev2le5anbg6oarljwh7lie" style="color: black;">Discrete Applied Mathematics</a> </i> &nbsp;
In this paper, we provide a construction of all the optimal graphs for the identification of vertices, that is to say graphs on n vertices having an identifying code of cardinality log 2 (n + 1) .  ...  Given a graph G on n vertices, it is easy to see that the minimum cardinality of an identifying code of G is at least log 2 (n + 1) .  ...  By construction, all the vertices of G(H ) have distinct and non-empty identifying sets, hence G(H ) is an optimal graph.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.dam.2006.03.011">doi:10.1016/j.dam.2006.03.011</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/pdvaognmx5gu3j2gmgj4n57tou">fatcat:pdvaognmx5gu3j2gmgj4n57tou</a> </span>
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On the size of identifying codes in triangle-free graphs

Florent Foucaud, Ralf Klasing, Adrian Kosowski, André Raspaud
<span title="">2012</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/lx7dev2le5anbg6oarljwh7lie" style="color: black;">Discrete Applied Mathematics</a> </i> &nbsp;
In an undirected graph G, a subset C⊆ V(G) such that C is a dominating set of G, and each vertex in V(G) is dominated by a distinct subset of vertices from C, is called an identifying code of G.  ...  The concept of identifying codes was introduced by Karpovsky, Chakrabarty and Levitin in 1998. For a given identifiable graph G, let (G) be the minimum cardinality of an identifying code in G.  ...  Given a graph G and a subset S of its vertices, we say that a set C ⊆ S is an S-identifying code of G if C is an identifying code of G [S] .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.dam.2012.02.009">doi:10.1016/j.dam.2012.02.009</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/hfkdtg5orbebbl5ezupn3a4d7i">fatcat:hfkdtg5orbebbl5ezupn3a4d7i</a> </span>
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On a new class of codes for identifying vertices in graphs

M.G. Karpovsky, K. Chakrabarty, L.B. Levitin
<span title="">1998</span> <i title="Institute of Electrical and Electronics Engineers (IEEE)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/niovmjummbcwdg4qshgzykkpfu" style="color: black;">IEEE Transactions on Information Theory</a> </i> &nbsp;
We also describe the identification of sets of vertices using covering codes that uniquely identify single vertices.  ...  We develop methods for constructing optimal topologies that yield identifying codes with a minimum number of codewords.  ...  ACKNOWLEDGMENT The authors wish to thank the anonymous reviewers whose careful reading and constructive comments have significantly improved the quality of the paper.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/18.661507">doi:10.1109/18.661507</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/fvu6se7p4nhzfklbajuhdiuxem">fatcat:fvu6se7p4nhzfklbajuhdiuxem</a> </span>
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Identifying Codes and Covering Problems

Moshe Laifenfeld, Ari Trachtenberg
<span title="">2008</span> <i title="Institute of Electrical and Electronics Engineers (IEEE)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/niovmjummbcwdg4qshgzykkpfu" style="color: black;">IEEE Transactions on Information Theory</a> </i> &nbsp;
Unlike identifying codes, every graph admits a trivial locating-dominating set -the entire set of vertices.  ...  The identifying code problem for a given graph involves finding a minimum set of vertices whose neighborhoods uniquely overlap at any given graph vertex.  ...  In the construction of G we use the well known fact [27] that an undirected graph of n vertices can be constructed with an identifying code of size ⌈lg n⌉ (assuming that an empty set is a valid identifying  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/tit.2008.928263">doi:10.1109/tit.2008.928263</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/onbllghh2jegjibyel6kahbbhq">fatcat:onbllghh2jegjibyel6kahbbhq</a> </span>
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The Complexity of the Identifying Code Problem in Restricted Graph Classes [chapter]

Florent Foucaud
<span title="">2013</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
An identifying code is a subset of vertices of a graph such that each vertex is uniquely determined by its nonempty neighbourhood within the identifying code.  ...  In particular, our work exhibits important graph classes for which Minimum Dominating Set is efficiently solvable, but Minimum Identifying Code is hard (whereas in all previously studied classes, their  ...  We now describe two constructions that ensure that the vertices of some vertex set A are correctly identified using the vertices of another set L. A over (A, L) ).  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-642-45278-9_14">doi:10.1007/978-3-642-45278-9_14</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/aw6j57tliret7lmzaxdbkkrhra">fatcat:aw6j57tliret7lmzaxdbkkrhra</a> </span>
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New results on variants of covering codes in Sierpiński graphs

Sylvain Gravier, Matjaž Kovše, Michel Mollard, Julien Moncel, Aline Parreau
<span title="2012-03-21">2012</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/c45m6ttnaje4xbjsq7m2c6df2a" style="color: black;">Designs, Codes and Cryptography</a> </i> &nbsp;
In this paper we study identifying codes, locating-dominating codes, and total-dominating codes in Sierpiński graphs. We compute the minimum size of such codes in Sierpiński graphs.  ...  Now, observe that if n = 2, then the set of inner vertices of S(2, k) is an identifying code of S(2, k), of cardinality k(k −1).  ...  It is an identifying code if it is a covering code of G that separates all pairs of distinct vertices of G.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s10623-012-9642-1">doi:10.1007/s10623-012-9642-1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/dt5u2uf2wbbdze3zxbatvutbta">fatcat:dt5u2uf2wbbdze3zxbatvutbta</a> </span>
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New results on variants of covering codes in Sierpinski graphs [article]

Sylvain Gravier , Julien Moncel
<span title="2012-01-05">2012</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
In this paper we study identifying codes, locating-dominating codes, and total-dominating codes in Sierpinski graphs. We compute the minimum size of such codes in Sierpinski graphs.  ...  Now, observe that if n = 2, then the set of inner vertices of S(2, k) is an identifying code of S(2, k), of cardinality k(k −1).  ...  It is an identifying code if it is a covering code of G that separates all pairs of distinct vertices of G.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1201.1202v1">arXiv:1201.1202v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/2blham6a4nglvcwpqnwta5pyxy">fatcat:2blham6a4nglvcwpqnwta5pyxy</a> </span>
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Decision and approximation complexity for identifying codes and locating-dominating sets in restricted graph classes

Florent Foucaud
<span title="">2015</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/usw2n4yaarcurchx7i4on2iqea" style="color: black;">Journal of Discrete Algorithms</a> </i> &nbsp;
When only vertices out of the code are asked to be identified, we get the related concept of a locating-dominating set.  ...  An identifying code is a subset of vertices of a graph with the property that each vertex is uniquely determined (identified) by its nonempty neighbourhood within the identifying code.  ...  We now describe two constructions, that ensure that the vertices of some vertex set A are correctly identified using the vertices of another set L.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.jda.2014.08.004">doi:10.1016/j.jda.2014.08.004</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/xne3ofpv5jf6xmq4lp56prvgzu">fatcat:xne3ofpv5jf6xmq4lp56prvgzu</a> </span>
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Locating-dominating sets and identifying codes in graphs of girth at least 5 [article]

Camino Balbuena, Florent Foucaud, Adriana Hansberg
<span title="2015-11-24">2015</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Locating-dominating sets and identifying codes are two closely related notions in the area of separating systems.  ...  In this paper, we study the size of a smallest locating-dominating set or identifying code for graphs of girth at least 5 and of given minimum degree.  ...  Identifying codes We now give constructions with large identifying code number. We start with a construction based on the 5-cycle C 5 , which has identifying code number 3 [4] . Proposition 25.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1407.7263v2">arXiv:1407.7263v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ntwcf6alazhafoc5jsntl3oc6a">fatcat:ntwcf6alazhafoc5jsntl3oc6a</a> </span>
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Minimizing the size of an identifying or locating-dominating code in a graph is NP-hard

Irène Charon, Olivier Hudry, Antoine Lobstein
<span title="">2003</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/elaf5sq7lfdxfdejhkqbtz6qoq" style="color: black;">Theoretical Computer Science</a> </i> &nbsp;
If the sets Br(v) ∩ C, v ∈ V (respectively, v ∈ V \C), are all nonempty and di erent, where Br(v) denotes the set of all points within distance r from v, we call C an r-identifying code (respectively,  ...  Let G = (V; E) be an undirected graph and C a subset of vertices.  ...  Two vertices having di erent identifying sets are said to be r-separated or separated. A code C is called r-identifying or identifying, if the sets IS r (v), v ∈ V , are all nonempty and di erent.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/s0304-3975(02)00536-4">doi:10.1016/s0304-3975(02)00536-4</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/oz2ag7s7ujda7jvihokfa6qyda">fatcat:oz2ag7s7ujda7jvihokfa6qyda</a> </span>
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The minimum identifying code graphs

André Raspaud, Li-Da Tong
<span title="">2012</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/lx7dev2le5anbg6oarljwh7lie" style="color: black;">Discrete Applied Mathematics</a> </i> &nbsp;
A set S of vertices is called an identifying code of G if, for every pair of distinct vertices u and v, both B(u) ∩ S and B(v) ∩ S are nonempty and distinct.  ...  A minimum identifying code of a graph G is an identifying code of G with minimum cardinality and M(G) is the cardinality of a minimum identifying code for G.  ...  Acknowledgments The authors thank the referee for many constructive suggestions.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.dam.2012.01.015">doi:10.1016/j.dam.2012.01.015</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ccyefe3u2na3fh3h5la4andopa">fatcat:ccyefe3u2na3fh3h5la4andopa</a> </span>
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Identifying and locating-dominating codes on chains and cycles

Nathalie Bertrand, Irène Charon, Olivier Hudry, Antoine Lobstein
<span title="">2004</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/54t3hgai4fhhthc74mj7z7tapu" style="color: black;">European journal of combinatorics (Print)</a> </i> &nbsp;
If for all vertices v ∈ V (respectively, v ∈ V \C), the sets B r (v) ∩ C are all nonempty and different, then we call C an r -identifying code (respectively, an r -locating-dominating code).  ...  Consider a connected undirected graph G = (V, E), a subset of vertices C ⊆ V , and an integer r ≥ 1; for any vertex v ∈ V , let B r (v) denote the ball of radius r centered at v, i.e., the set of all vertices  ...  to the same set of segments (identifying codes) or no two noncodewords belong to the same set of segments (locating-dominating codes).  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.ejc.2003.12.013">doi:10.1016/j.ejc.2003.12.013</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/q3hhk5sjrzeodcr5jpxb3l7h5q">fatcat:q3hhk5sjrzeodcr5jpxb3l7h5q</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20061210024748/http://www.lsv.ens-cachan.fr/Publis/PAPERS/PDF/BCHL-ejc04.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/b3/de/b3dec976660f4caab60a11622df824313be5257c.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.ejc.2003.12.013"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> elsevier.com </button> </a>

Locating-Dominating Sets and Identifying Codes in Graphs of Girth at least 5

Camino Balbuena, Florent Foucaud, Adriana Hansberg
<span title="2015-04-29">2015</span> <i title="The Electronic Journal of Combinatorics"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/v5dyak6ulffqfara7hmuchh24a" style="color: black;">Electronic Journal of Combinatorics</a> </i> &nbsp;
Locating-dominating sets and identifying codes are two closely related notions in the area of separating systems.  ...  In this paper, we study the size of a smallest locating-dominating set or identifying code for graphs of girth at least 5 and of given minimum degree.  ...  The black vertices form an optimal identifying code and locating-dominating set of P 10 .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.37236/4562">doi:10.37236/4562</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/zl43vidvrvbn3hcru735qlgoe4">fatcat:zl43vidvrvbn3hcru735qlgoe4</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20171225141803/http://www.combinatorics.org:80/ojs/index.php/eljc/article/download/v22i2p15/pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/8d/b9/8db980b7bc567e667ad0bf84b61b96479d44b762.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.37236/4562"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> Publisher / doi.org </button> </a>

Locating and Identifying Codes in Circulant Networks [article]

M. Ghebleh, L. Niepel
<span title="2012-07-19">2012</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
A set S of vertices of a graph G is a dominating set of G if every vertex u of G is either in S or it has a neighbour in S.  ...  A set S ⊆ V(G) is called an identifying code in G, if the sets S∩ N[u] where u∈ V(G) are all nonempty and distinct. We study locating and identifying codes in the circulant networks C_n(1,3).  ...  The sets B t can indeed be used in constructions of identifying codes for the graphs C n (1, 3) when n is not necessarily a multiple of 11.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1207.4660v1">arXiv:1207.4660v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/fxdwps75dbebpot2gbfw7xs2ky">fatcat:fxdwps75dbebpot2gbfw7xs2ky</a> </span>
<a target="_blank" rel="noopener" href="https://archive.org/download/arxiv-1207.4660/1207.4660.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> File Archive [PDF] </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1207.4660v1" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>
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