Open-independent, open-locating-dominating sets

Abstract

A distinguishing set for a graph G = (V, E) is a dominating set D, each vertex v ∈ D being the location of some form of a locating device, from which one can detect and precisely identify any given "intruder" vertex in V (G). As with many applications of dominating sets, the set D might be required to have a certain property for D , the subgraph induced by D (such as independence, paired, or connected). Recently the study of independent locating-dominating sets and independent identifying codes was initiated. Here we introduce the property of open-independence for openlocating-dominating sets.

Open-independent, open-locating-dominating sets: structural aspects of some classes of graphs

“…Open-independent, open-locating-dominating sets were first studied by Seo and Slater [33] who stated necessary and sufficient conditions for the existence of an OLD oind -set in a graph G that has girth g(G) ≥ 5.…”

“…For general graphs (with arbitrary girth), the conditions stated in Theorem 2. Seo and Slater [33] presented some results about OLD oind -sets in trees. The authors showed that every leaf and its neighbor are contained in any OLD oind -set of any tree T , if T has such a set.…”

“…The reader is referred to [26] for an on-line bibliography on this topic and related notions. In this paper we consider the following more restrictive problem: To the best of our knowledge, Seo and Slater [33] were the first to study open-independent, open-locatingdominating sets (OLD oind -sets for short). They presented some results on OLD oind -sets in paths, trees and infinite grid graphs, and characterized OLD oind -sets in graphs with girth at least five.…”

Degree equitable restrained double domination in graphs

“…In societies, persons with nearly equal statuses tend to be friendly. For more details on the domination refer [6,7,10,12].…”

The geodetic domination number of comb product graphs

“…There are several modifications on domination concept in graph. Some of them are locatingdominating set [2,6,19,23], independent dominating set [4,14], Roman dominating set [9,13]. In this paper, we are interested to study another variant of domination in graph, namely geodeticdominating set.…”