On global location-domination in graphs

Abstract

A dominating set S of a graph G is called locating-dominating, LD-set for short, if every vertex v not in S is uniquely determined by the set of neighbors of v belonging to S . Locating-dominating sets of minimum cardinality are called LD-codes and the cardinality of an LD-code is the location-domination number λ(G). An LD-set S of a graph G is global if it is an LD-set of both G and its complement G. The global location-domination number λ g (G) is the minimum cardinality of a global LD-set of G. In this work, we give some relations between locating-dominating sets and the location-domination number in a graph and its complement.

The adjacency dimension of graphs

“…, is the minimum cardinality among all locating-dominating sets in G. A locating-dominating set of cardinality λ(G) is called a λ(G)-set. The concept of a locating-dominating set was introduced and first studied by Slater [24,25] and studied, for instance, in [4,14,19] and elsewhere.…”

LD-graphs and global location-domination in bipartite graphs

“…In this work, we carry out a similar study for bipartite graphs. After noticing that solving the equality λ(G) = λ(G) + 1 is closely related to analyzing the existence or not of sets that are simultaneously locating-dominating sets in both G and its complement G, the following definitions were introduced in [10]. Definition 1 ([10]).…”

“…This section is devoted to approach the relationship between λ(G) and λ(G), for any arbitrary graph G. Some of the results we present were previously shown in [9,10] and we include them for the sake of completeness.…”

“…We intend to obtain either necessary or sufficient conditions for a graph G to satisfy λ(G) > λ(G), i.e., λ(G) = λ(G) + 1. This problem was approached and completely solved in [10] for the family of block-cactus. In this work, we carry out a similar study for bipartite graphs.…”

Locating domination in bipartite graphs and their complements

“…(a) There is at most one vertex u ∈ V \ S such that N (u) ∩ S = S, and in the case it exists, S ∪ {u} is an LD-set of G. According to the preceding inequality, λ(G) ∈ {λ(G) − 1, λ(G), λ(G) + 1} for every graph G, all cases being feasible for some connected graph G. We intend to determine graphs such that λ(G) > λ(G), that is, we want to solve the equation λ(G) = λ(G) + 1. This problem was completely solved in [9] for the family of block-cactus.…”