On Graphs Whose Graphoidal Length Is Half of Its Size

Abstract

Let $G=(V,E)$ be a finite graph. A graphoidal cover $\varPsi$ of G is a collection of paths (not necessary open) in G such that every vertex of G is an internal vertex of at most one path in $\varPsi$ and every edge of G is in exactly one path in $\varPsi .$ The graphoidal covering number $\eta$ of G is the minimum cardinality of a graphoidal cover of G. The length $gl_{\varPsi }(G)$ of a graphoidal cover $\varPsi$ of G is defined to be $\min \{l(P): P\in \varPsi \}$ where l(P) is the length of the path P. The graphoidal length gl(G) is defined to be $\max \{gl_{\varPsi }(G): \varPsi$ is a graphoidal cover of $G\}.$ For any graph G of size q, $gl(G)\le q$ and this bound is attained if and only if G is either a path or a cycle. Further if $gl(G)\ne q$, then $gl(G)\le \left\lfloor q/2\right\rfloor$. In this paper we characterize graphs having graphoidal length $\left\lfloor q/2\right\rfloor$. In the process we obtain that there are exactly 12 non homomorphic graphs having graphoidal covering number two.

R. Singh is thankful to University Grants Commission (UGC) for providing research grant Schs/SRF/AA/139/F-212/2013-14/438.