A graphoidal cover of a given graph G=(V,E) is a set of its paths of length at least one, not necessarily open, such that no two paths have a common internal vertex and every edge of G is in exactly one of these paths. Graphoidal covers provide a fresh ground for generalizing results in graph theory and this paper is the first attempt to demonstrate the fruitfulness of this contention taking the notion of domination in graphs. Given a graphoidal cover ψ of G we define a set D of vertices of G to be a ψ-dominating set (ψ-domset, for short) of G whenever for every vertex v in V⧹D there exists a vertex u in D and a path P in ψ such that u and v are the end-vertices of P. This paper initiates a study of this concept in graphs which may not be necessarily finite.
