In 1998 Teresa Haynes, Stephen Hedetniemi, and Peter Slater [85, 86] wrote the two so-
called “domination books” and in so doing provided the first comprehensive treatment of …

Given a set of vertices S={v 1, v 2,…, vk} of a connected graph G, the metric representation of
a vertex v of G with respect to S is the vector r (v| S)=(d (v, v 1), d (v, v 2),…, d (v, vk)), where …

As a generalization of the concept of a metric basis, this article introduces the notion of $ k $-
metric basis in graphs. Given a connected graph $ G=(V, E) $, a set $ S\subseteq V $ is said …

For a graph G that models a facility, various detection devices can be placed at the vertices
so as to identify the location of an intruder such as a thief or saboteur. Here we introduce the …

Let G be a connected graph. A vertex w strongly resolves a pair u, v of vertices of G if there
exists some shortest u− w path containing v or some shortest v− w path containing u. A set W …

The problems of determining locating-dominating, open locating-dominating or locating total-
dominating sets of minimum cardinality in a graph G are variations of the classical minimum …

Let G=(V, E) be a connected graph. The distance between two vertices u, v∈ V, denoted by
d (u, v), is the length of a shortest u− v path in G. The distance between a vertex v∈ V and a …

In this paper, we continue the study of locating-total domination in graphs. A set S of vertices
of a graph G is a total dominating set of G if every vertex of G is adjacent to a vertex in S. We …

A vertex v ∈ V (G) v∈ V (G) is said to distinguish two vertices x, y ∈ V (G) x, y∈ V (G) of a
non-trivial connected graph G if the distance from v to x is different from the distance from v …

A set of vertices in a graph is (a) a dominating set if every vertex of Ò has a neighbor in,(b) a
locating-dominating set if for every two vertices Ù Ú of Ò the sets Æ (Ù) and Æ (Ú) are non …