Graphoidally independent infinite graphs

Abstract

A graphoidal cover of a graph G (not necessarily finite) is a collection ψ of paths in G, called ψ-edges, (not necessarily finite, not necessarily open) satisfying the following axioms: (GC-1) Every vertex of G is an internal vertex of at most one path in ψ, and (GC-2) every edge of G is in exactly one path in ψ. The pair is called a graphoidally covered graph. In a graphoidally covered graph two distinct vertices u and v of G are ψ-adjacent if they are the ends of a finite open path in ψ. A graphoidally covered graph (or G) in which no two distinct vertices are ψ-adjacent is called ψ-independent and a graph G possessing a graphoidal cover ψ such that G is ψ-independent is called a graphoidally independent graph. This paper is an attempt to characterize graphoidally independent infinite graphs. In this paper we establish complete characterization of graphoidally independent infinite trees, infinite unicyclic graphs and infinite 2-edge connected graphs.

Keywords

• graphoidal cover of a graph
• graphoidally covered graphs
• graphoidally independent graphs