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.
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Gupta, P., Singh, R., Arumugam, S. (2017). Graphoidal Length and Graphoidal Covering Number of a Graph. In: Arumugam, S., Bagga, J., Beineke, L., Panda, B. (eds) Theoretical Computer Science and Discrete Mathematics. ICTCSDM 2016. Lecture Notes in Computer Science(), vol 10398. Springer, Cham. https://doi.org/10.1007/978-3-319-64419-6_39
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