On graphs whose graphoidal domination number is one

Given a graph G=(V,E)G=(V,E), a set ψψ of non-trivial paths, which are not necessarily open, called ψψ-edges, is called a graphoidal cover of  GG if it satisfies the following conditions: (GC−1)(GC−1) Every vertex of GG is an internal vertex of at most one path in ψψ, and (GC−2)(GC−2) every edge of GG is in exactly one path in ψψ; the ordered pair (G,ψ)(G,ψ) is called a graphoidally covered graph. Two vertices u and v   of GG are ψψ-adjacent   if they are the ends of an open ψψ-edge. A set DD of vertices in (G,ψ)(G,ψ) is ψψ-dominating (in short  ψψ-dom set)   if every vertex of GG is either in DD or is ψψ-adjacent to a vertex in DD. Let γψ(G)=inf{|D|:Disaψ−domsetofG}γψ(G)=inf{|D|:Disaψ−domsetofG}. A ψψ-dom set DD with |D|=γψ(G)|D|=γψ(G) is called a γψ(G)γψ(G)-set. The graphoidal domination number of a graph  GG denoted by γψ0(G) is defined as   inf{γψ(G):ψ∈GG}inf{γψ(G):ψ∈GG}. Let GG be a connected graph with cyclomatic number μ(G)=(q−p+1)μ(G)=(q−p+1). In this paper, we characterize graphs for which there exists a non-trivial graphoidal cover ψψ such that γψ(G)=1γψ(G)=1 and l(P)>1l(P)>1 for each P∈ψP∈ψ and in this process we prove that the only such graphoidal covers are such that l(P)=2l(P)=2 for each P∈ψP∈ψ.