Graphoidal Length and Graphoidal Covering Number of a Graph

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 $G\}.$ In this paper we investigate the existence of graphs which admit a graphoidal cover $\varPsi$ with $|\varPsi |=\eta (G)$ and $gl_{\varPsi }(G)=gl(G)$.

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