Bounds on Graphoidal Length of a Graph

Abstract A graphoidal cover of a graph G is a set Ψ of non-trivial paths (which are not necessarily open) in G such that every vertex of G is an internal vertex of at most one path in Ψ and every edge of G is in exactly one path in Ψ. We denote the set of all graphoidal covers of graph G by G G . In this paper we introduce a parameter gl(G), called graphoidal length of the graph G and is defined as g l ( G ) = max Ψ ∈ G G ⁡ { min P ∈ Ψ ⁡ l ( P ) } . We give bounds for the parameter gl(G) in terms of the well known and well studied parameter η ( G ) , graphoidal covering number of the graph and show that the bounds are sharp.