Bounds on co-independent liar's domination in graphs

Abstract

A set S⊆V of a graph G = (V, E) is called a co-independent liar's dominating set of G if (i) for all v ∈ V, |NG[v] ∩ S| ≥ 2, (ii) for every pair u, v ∈ V of distinct vertices, |(NG[u] ∪ NG[v]) ∩ S| ≥ 3, and (iii) the induced subgraph of G on V − S has no edge. The minimum cardinality of vertices in such a set is called the co-independent liar's domination number of G, and it is denoted by cLRcoi(G). In this paper, we introduce the concept of co-independent liar's domination number of the middle graph of some standard graphs such as path and cycle graphs, and we propose some bounds on this new parameter.