$1$-Movable Clique Dominating Sets of a Graph

Abstract

A clique (convex) dominating set S of G is a 1-movable clique dominating set (resp. 1-movable convex dominating set) of G if for every v ∈ S, either S \ {v} is a clique (resp. convex) dominating set or there exists a vertex u ∈ (V (G) \ S) ∩ NG(v) such that (S \ {v}) ∪ {u} is a clique (resp. convex) dominating set of G. The minimum cardinality of a 1-movable clique (resp. 1-movable convex) dominating set of G, denoted by γ 1 mcl (G) (resp. γ 1 mcon (G)), is called the 1-movable clique domination number (resp. 1-movable convex domination number) of G. A 1-movable clique dominating set in G with cardinality γ 1 mcl (G) is called a γ 1 mcl -set of G. This paper aims to characterize the 1-movable clique dominating sets of some graphs including those resulting from the join and composition of two graphs. The corresponding 1-movable clique domination number of the resulting graph is then determined. Further, it is shown that the concepts of 1-movable clique domination and 1-movable convex domination are equivalent.

Cliques and Supercliques in Graphs

“…Other studies on domination are in [11], [12], [17], and [22]. Some studies involving cliques can be found in [5], [6], [7], [8], [13], [15], [18], and [23].…”