Identifying Codes in Line Graphs

Foucaud, Florent; Gravier, Sylvain; Naserasr, Reza; Parreau, Aline; Valicov, Petru

doi:10.1002/jgt.21686

Cited by 30 publications

(33 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We refer to e.g. [1,12,16,17,18,19,21,29,31] for some results. On the positive side, Identifying Code, Locating-Dominating-Set and Open Locating-Dominating Set are linear-time solvable for graphs of bounded clique-width (using Courcelle's theorem [15] ).…”

Section: Similarly a Graph Admits An Open Locating-dominating Set Ifmentioning

confidence: 99%

Identification, location–domination and metric dimension on interval and permutation graphs. I. Bounds

Foucaud

Mertzios

Naserasr

et al. 2017

Theoretical Computer Science

Self Cite

View full text Add to dashboard Cite

We consider the problems of finding optimal identifying codes, (open) locating-dominating sets and resolving sets of an interval or a permutation graph. In these problems, one asks to find a subset of vertices, normally called a solution set, using which all vertices of the graph are distinguished. The identification can be done by considering the neighborhood within the solution set, or by employing the distances to the solution vertices. Normally the goal is to minimize the size of the solution set then. Here we study the case of interval graphs, unit interval graphs, (bipartite) permutation graphs and cographs. For these classes of graphs we give tight lower bounds for the size of such solution sets depending on the order of the input graph. While such lower bounds for the general class of graphs are in logarithmic order, the improved bounds in these special classes are of the order of either quadratic root or linear in terms of number of vertices. Moreover, the results for cographs lead to linear-time algorithms to solve the considered problems on inputs that are cographs. * u, v if it (totally) dominates exactly one of them. A set S (totally) separates the vertices of a set X if all pairs of X are (totally) separated by a vertex of S. Whenever it is clear from the context, we will only say "separate" and omit the word "totally". We have the three key definitions, that merge the concepts of (total) domination and (total) separation:Definition 1 (Slater [33,34]). A set S of vertices of a graph G is a locating-dominating set if it is a dominating set and it separates the vertices of V (G) \ S. The smallest size of a locating-dominating set of G is the location-domination number of G, denoted γ LD (G). Without the domination constraint, this concept has also been used under the name distinguishing set in [2] and sieve in [28].Definition 2 (Karpovsky, Chakrabarty and Levitin [27]). A set S of vertices of a graph G is an identifying code if it is a dominating set and it separates all vertices of V (G). The smallest size of an identifying code of G is the identifying code number of G, denoted γ ID (G). Definition 3 (Seo and Slater [31]). A set S of vertices of a graph G is an open locating-dominating set if it is a total dominating set and it totally separates all vertices of V (G). The smallest size of an open locating-dominating set of G is the open location-domination number of G, denoted γ OLD (G). This concept has also been called identifying open code in [25]. Separation could also be done using distances from the members of the solution set. Let d(x, u) denote the distance between two vertices x and u. Definition 4 (Harary and Melter [24], Slater [32]). A set B of vertices of a graph G is a resolving set if for each pair u, v of distinct vertices, there is a vertex x of B with d(x, u) = d(x, v). 1 The smallest size of a resolving set of G is the metric dimension of G, denoted dim(G).

show abstract

Section: Similarly a Graph Admits An Open Locating-dominating Set Ifmentioning

confidence: 99%

Identification, location–domination and metric dimension on interval and permutation graphs. I. Bounds

Foucaud

Mertzios

Naserasr

et al. 2017

Theoretical Computer Science

Self Cite

View full text Add to dashboard Cite

show abstract

“…On the positive side, Locating-Dominating Set is constant-factor approximable for bounded degree graphs [20], line graphs [14,15], interval graphs [4] and is linear-time solvable for graphs of bounded clique-width (using Courcelle's theorem [8]). Furthermore, an explicit linear-time algorithm solving LocatingDominating Set on trees is known [33].…”

Section: Metric Dimensionmentioning

confidence: 99%

Algorithms and Complexity for Metric Dimension and Location-domination on Interval and Permutation Graphs

Foucaud

Mertzios

Naserasr

et al. 2016

Graph-Theoretic Concepts in Computer Science

Self Cite

View full text Add to dashboard Cite

Abstract. We study the problems Locating-Dominating Set and Metric Dimension, which consist of determining a minimum-size set of vertices that distinguishes the vertices of a graph using either neighbourhoods or distances. We consider these problems when restricted to interval graphs and permutation graphs. We prove that both decision problems are NP-complete, even for graphs that are at the same time interval graphs and permutation graphs and have diameter 2. While Locating-Dominating Set parameterized by solution size is trivially xed-parameter-tractable, it is known that Metric Dimension is W [2]-hard. We show that for interval graphs, this parameterization of Metric Dimension is xed-parameter-tractable.

show abstract

“…On the positive side, there always exists a O(log |V (G)|) approximation for Min Id Code [33]. Moreover, even if in the general case Min Id Code is hard to evaluate, there exist several constant approximation algorithms for restricted classes of graphs, such as planar graphs [29] or line graphs [15].…”

mentioning

confidence: 99%

Identifying Codes in Hereditary Classes of Graphs and VC-Dimension

Bousquet¹,

Lagoutte²,

Li³

et al. 2015

SIAM J. Discrete Math.

Self Cite

View full text Add to dashboard Cite

Abstract. An identifying code of a graph is a subset of its vertices such that every vertex of the graph is uniquely identified by the set of its neighbours within the code. We show a dichotomy for the size of the smallest identifying code in classes of graphs closed under induced subgraphs. Our dichotomy is derived from the VC-dimension of the considered class C, that is the maximum VC-dimension over the hypergraphs formed by the closed neighbourhoods of elements of C. We show that hereditary classes with infinite VC-dimension have infinitely many graphs with an identifying code of size logarithmic in the number of vertices while classes with finite VC-dimension have a polynomial lower bound.We then turn to approximation algorithms. We show that Min Id Code (the problem of finding a smallest identifying code in a given graph from some class C) is log-APX-hard for any hereditary class of infinite VCdimension. For hereditary classes of finite VC-dimension, the only known previous results show that we can approximate Min Id Code within a constant factor in some particular classes, e.g. line graphs, planar graphs and unit interval graphs. We prove that Min Id Code can be approximate within a factor 6 for interval graphs. In contrast, we show that Min Id Code on C 4 -free bipartite graphs (a class of finite VC-dimension) cannot be approximated to within a factor of c log(|V |) for some c > 0. [19]. For a complete survey on these results, the reader is referred to the online bibliography of Lobstein [24].Two vertices u and v are twins if. The whole vertex set V (G) is an identifying code if and only if G is twin-free. Since supersets of identifying codes are identifying, an identifying code exists for G if and only if it is twin-free. A natural problem in the study of identifying codes is to find one of a minimum size. Given a twin-free graph G, the smallest size of an identifying code of G is called the identifying code number of G and is denoted by γ ID (G). The problem of determining γ ID is called the Min Id Code problem, and its decision version is NP-complete [8]. Let X ⊆ V . We denote by G[X] the graph induced by the subset of vertices X. In this paper, we focus on hereditary classes of graphs, that is classes closed under taking induced subgraphs. We consider the two following problems: finding good lower bounds and approximation algorithms for the identifying code number.

show abstract

Identifying Codes in Line Graphs

Cited by 30 publications

References 15 publications

Identification, location–domination and metric dimension on interval and permutation graphs. I. Bounds

Identification, location–domination and metric dimension on interval and permutation graphs. I. Bounds

Algorithms and Complexity for Metric Dimension and Location-domination on Interval and Permutation Graphs

Identifying Codes in Hereditary Classes of Graphs and VC-Dimension

Contact Info

Product

Resources

About