The maximum number of short paths in a Halin graph

Abstract

A Halin graph G is a plane graph consisting of a plane embedding of a tree T of order at least 4 containing no vertex of degree 2, and of a cycle C connecting all leaves of T. Let fh(n,G) be the maximum number of copies of G in a Halin graph on n vertices. In this paper, we give exact values of fh(n,G) when G is a path on k vertices for 2≤k≤5. Moreover, we develop a new graph transformation preserving the number of vertices, so that the resulting graph has a monotone behavior with respect to the number of short paths.