Bounds on Graphoidal Length of a Graph

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 GGGG. In this paper we introduce a parameter gl(G), called graphoidal length of the graph G   and is defined as gl(G)=maxΨ∈GG⁡{minP∈Ψ⁡l(P)}gl(G)=maxΨ∈GG⁡{minP∈Ψ⁡l(P)}. We give bounds for the parameter gl(G  ) in terms of the well known and well studied parameter η(G)η(G), graphoidal covering number of the graph and show that the bounds are sharp.