LD-graphs and global location-domination in bipartite graphs

Abstract

Abstract.A dominating set S of a graph G is a locating-dominating-set, 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 LDcodes 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 for both G and its complement, G. One of the main contributions of this work is the definition of the LD-graph, an edge-labeled graph associated to an LD-set, that will be very helpful to deduce some properties of location-domination in graphs. Concretely, we use LD-graphs to study the relation between the location-domination number in a bipartite graph and its complement.

Locating-dominating number of certain infinite families of convex polytopes with applications

“…In this situation, if for each and the sum of weights for closed neighborhoods is not less than 1, then D is a dominating set of Γ, we denote a dominating set by S . If, for all distinct vertices , it holds that , then the dominating set is called a locating-dominating set (see Hernando et al [1] ). The minimum cardinality of a locating-dominating set is denoted by , which stands for the locating-dominating number of Γ.…”

Binary Locating-Dominating Sets in Rotationally-Symmetric Convex Polytopes

“…The concepts of fault-tolerant locating-dominating and open neighborhood locating-dominating sets in trees have been studied by Seo et al [6,7] and Salter [8]. For more on locating-dominating sets and related parameters, we suggest the reader to [5,[9][10][11].…”

On global location-domination in graphs

“…• In this work, we have completely solved the equality λ(G) = λ(G) + 1 for the block-cactus family. In [13], a similar study has been done for the family of bipartite graphs. We suggest to approach this problem for other families of graphs, such as outerplanar graphs, chordal graphs and cographs.…”