Acyclic Distance Closed Domination Critical Graphs

Abstract

In a graph $G = (V, E)$, a set $S\subset V(G)$ is said to be an acyclic distance closed dominating set if (i) $\langle S\rangle$ is distance closed and (ii) $\langle S\rangle$ is acyclic. The cardinality of the minimum acyclic distance closed dominating set is called an acyclic distance closed domination number and it is denoted by $\gamma_{adcl}(G)$. In this paper, we discuss the critical concept in acyclic distance closed domination which deals with those graphs that are critical in the sense that their acyclic distance closed domination number drops when any missing edge is added. Also, we analyze some structural properties of those acyclic distance closed domination critical graphs.