Changing of the Number of Minimum Dominating Sets after Edge Addition: Non Critical Edges

Let γ(G) and #γ(G) denote the domination number and the number of all distinct minimum dominating sets of a graph G, respectively. We show that #γ(G+ e) ≥ #γ(G) for every edge e ∈ E(G) with γ(G+e) = γ(G). Based on this, we begin an investigation of graphs for which #γ increases whenever an edge is added (we call such graphs #γ-EA-critical). We prove that if G is #γ-EA-critical then either G is edgeless or γ(G + e) = γ(G) for all e ∈ E(G). If p is a number of endvertices of a #γ-EA-critical graph G of order n ≥ 6, we show that (a) p ≤ γ(G), and (b) p ≤ n/3 provided G is connected. In both cases, we find the extremal graphs. We also find all #γ-EA-critical graphs each component of which is either a tree or a unicyclic graph.