Locating–dominating sets in twin-free graphs

Abstract

a b s t r a c tA locating-dominating set of a graph G is a dominating set D of G with the additional property that every two distinct vertices outside D have distinct neighbors in D; that is, for distinct vertices u and v outside D, N(u) denotes the open neighborhood of u. A graph is twin-free if every two distinct vertices have distinct open and closed neighborhoods. The location-domination number of G, denoted γ L (G), is the minimum cardinality of a locating-dominating set in G. It is conjectured by Garijo et al. (2014) that if G is a twin-free graph of order n without isolated vertices, then γ L (G) ≤ n 2 .We prove the general bound γ L (G) ≤ 2n 3 , slightly improving over the ⌊ 2n 3 ⌋ + 1 bound of Garijo et al. We then provide constructions of graphs reaching the n 2 bound, showing that if the conjecture is true, the family of extremal graphs is a very rich one. Moreover, we characterize the trees G that are extremal for this bound. We finally prove the conjecture for split graphs and co-bipartite graphs.

Related Wheel Graphs and Its Locating Edge Domination Number

“…Dominating set has various applications in real life such as in network topology, transportation system, facility location problems, and several other fields [3,7,8,12]. The locating edge dominating set is one of a new study in dominating set topic.…”

Domination and location in twin-free digraphs

“…In this section, we propose a general method to obtain locating-dominating sets of twin-free digraphs, based on special dominating sets. This method was used in [6] for the case of undirected graphs (a similar argument was also used in [9]). It was adapted to digraphs in [12] for quasi-twin-free digraphs, and here we extend it to all twin-free digraphs.…”

Location-domination in line graphs

“…This bound is tight and the extremal examples have been classified, see [23]. As observed in [13], while there are many graphs (without isolated vertices) which have location-domination number much larger than one-half their order, the only such graphs that are known contain many twins. For example, for the complete graph K n of order n, we have γ L (K n ) = n − 1 for all n ≥ 3.…”

“…Among the existing variations of (total) domination, the one of location-domination and locationtotal domination are widely studied. A set D of vertices locates a vertex v / ∈ D if the neighborhood of v within D is unique among all vertices in V (G) \ D. A locating-dominating set is a dominating set D that locates all the vertices in V (G) \ D, and the location-domination number of G, denoted γ L (G), is the minimum cardinality of a locating-dominating set in G. A locating-total dominating set, abbreviated LTD-set, is a TD-set D that locates all the vertices, and the location-total domination number of G, denoted γ L t (G), is the minimum cardinality of a LTD-set in G. The concept of a locating-dominating set was introduced and first studied by Slater [26,27] (see also [9,10,13,25,28]), and the additional condition that the locating-dominating set be a total dominating set was first considered in [18] (see also [1,2,3,5,6,7,19,20]).…”

“…Theorem ( [12,13,15]) The statement of Conjecture 2 is true if the twin-free graph G of order n (without isolated vertices) satisfies any of the following conditions.…”