A graph is said to be well-edge-dominated if all its minimal edge dominating sets are minimum. It known that every G also equimatchable, meaning maximal matching in maximum. In this paper, we show a connected, triangle-free, nonbipartite, graph, then one of three graphs. We characterize the split graphs and Cartesian products. particular, connected product $$G\Box H$$ well-edge-dominated, where...

We disprove a conjecture by Skupień that every tree of order n has at most 2 minimal dominating sets. We construct a family of trees of both parities of the order for which the number of minimal dominating sets exceeds 1.4167. We also provide an algorithm for listing all minimal dominating sets of a tree in time O(1.4656). This implies that every tree has at most 1.4656 minimal dominating sets.

For a given graph G = (V,E), a set D ⊆ V (G) is said to be an outerconnected dominating set if D is dominating and the graph G−D is connected. The outer-connected domination number of a graph G, denoted by γ̃c(G), is the cardinality of a minimum outer-connected dominating set of G. We study several properties of outer-connected dominating sets and give some bounds on the outer-connected dominati...

Domination theory is an important branch of Graph Theory and has many applications in Engineering, Communication Networks and many others. Allan, R.B., and Laskar, R., [1,2], Cockayne, E.J.,and Hedetniemi, S.T., [3], Haynes, T.W., and Slater, J.S., [4], have studied various domination parameters of graphs. Graphs associated with certain arithmetic functions which are usually called arithmetic g...

Given a graphG, the k-dominating graph ofG, Dk(G), is defined to be the graph whose vertices correspond to the dominating sets of G that have cardinality at most k. Two vertices in Dk(G) are adjacent if and only if the corresponding dominating sets of G differ by either adding or deleting a single vertex. The graph Dk(G) aids in studying the reconfiguration problem for dominating sets. In parti...