On 2 acyclic simple graphoidal covering of bicyclic graphs

Abstract

A 2−simple graphoidal cover of G is a set ψ of (not necessarily open) paths in G such that every edge is in exactly one path in ψ and every vertex is an internal vertex of at most two paths in ψ and any two paths in ψ has at most one vertex in common. The minimum cardinality of the 2−simple graphoidal cover of G is called the 2−simple graphoidal covering number of G and is denoted by η2s. A 2−simple graphoidal cover ψ of a graph G is called 2−acyclic simple graphoidal cover if every member of ψ is a path. The minimum cardinality of a 2−acyclic simple graphoidal cover of G is called the 2−acyclic graphoidal covering number of G and is denoted by η2as. This paper discusses 2−acyclic simple graphoidal cover on bicyclic graphs.