FURTHER RESULTS ON DOMINATION IN GRAPHOIDALLY COVERED GRAPHS

In any graph G = (V, E) that is not necessarily finite, a graphoidal cover is a set. of nontrivial paths P-1, P-2,..., not necessarily open and called. psi -edges, such that (GC- 1) no vertex of G is an internal vertex of more than one path in psi, and (GC- 2) every edge of G is in exactly one of the paths in psi. A psi -dominating set of G is then defined as a set D of vertices in G such that every vertex of G is either in D or is an end-vertex of a psi -edge having its other end-vertex in D. In this note, we present some new results that facilitate having more insight into the notion of psi -domination in graphs; particularly, we give a characterization of (i) finite connected graphs possessing a graphoidal cover psi such that (G, psi) is psi -independent and (ii) trees and unicyclic graphs which possess a graphoidal cover. such that their psi -domination numbers turn out to be one.