Domination in graphoidally covered graphs: Least-kernel graphoidal graphs-II

This paper is based on a conjecture by Dr. B.D Acharya, “Every graph possesses an LKG cover”. After finding an example of a graph which does not possess an LKG cover, we obtain a necessary condition in the form of forbidden subgraph for a graph to be a least kernel graphoidal graph. We further prove that the condition is sufficient for a block graph with a unique nontrivial block. Thereafter we identify certain classes of graphs in which every graph possesses an LKG cover. Moreover, following our surmise that every graph with Δ≤6 possesses an LKG cover, we were able to show that every finite graph with Δ≤3 is indeed an LKG graph.