A Note on the Characterization of Potentially \(K_{1,1,s}\)-Graphic Sequences

Abstract

For a given graph \(H\), a graphic sequence \(\pi = (d_1, d_2, \ldots, d_n)\) is said to be potentially \(H\)-graphic if there is a realization of \(\pi\) containing \(H\) as a subgraph. In this paper, we characterize potentially \(K_{1,1,6}\)-positive graphic sequences. This characterization implies the value of \(\sigma(K_{1,1,6}, n)\). Moreover, we also give a simple sufficient condition for a positive graphic sequence \(\pi = (d_1, d_2, \ldots, d_n)\) to be potentially \(K_{1,1,s}\)-graphic for \(n \geq s+2\) and \(s \geq 2\).