Spanning Trees of \(K_{1,4}\)-free Graphs with a Bounded Number of Leaves and Branch Vertices

Abstract

Let T be a tree. A vertex of degree one is a leaf of T and a vertex of degree at least three is a branch vertex of T. A graph is said to be \(K_{1,4}\)-free if it does not contain \(K_{1,4}\) as an induced subgraph. In this paper, we study the spanning trees with a bounded number of leaves and branch vertices of \(K_ {1,4}\)-free graphs. Applying the main results, we also give some improvements of previous results on the spanning tree with few branch vertices for the case of \(K_{1,4}\)-free graphs.