Further Results on Domination in Graphoidally Covered Graphs

Abstract

In any graph G = (V, E) that is not necessarily finite, a graphoidal cover is a set ψ of nontrivial paths P1, P,…, not necessarily open and called ψ-edges, such that (GC-1) no vertex of G is an internal vertex of more than one path in ψ, and (GC-2) every edge of G is in exactly one of the paths in ψ. A ψ -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 ψ -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 ψ -domination in graphs; particularly, we give a characterization of (i) finite connected graphs possessing a graphoidal cover ψ such that (G, ψ) is ψ -independent and (ii) trees and unicyclic graphs which possess a graphoidal cover ψ such that their ψ -domination numbers turn out to be one.