On graphoidal length of a tree in terms of its diameter

• Main Authors: Purnima Gupta, Megha Agarwal, Rajesh Singh
• Format: Article
• Language: English
• Published: Taylor & Francis Group 2020-09-01
• Series: AKCE International Journal of Graphs and Combinatorics
• Subjects: graphoidal cover; graphoidally covered graph; graphoidal length; graphoidal covering number
• Online Access: http://dx.doi.org/10.1016/j.akcej.2019.12.012

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 The graphoidal length gl(G) of a graph G is defined as In this paper, we obtain bounds for the graphoidal length of a tree in terms of its diameter. We prove that if G is any tree (excepts paths) of diameter d, then graphoidal length gl(G) is less than equal to Further, we characterize trees attaining the upper bound. Also, the trees for which gl(G) = k where are characterized.