Degree Sequences of Halin Graphs, and Forcibly Cograph-graphic Sequences

Abstract

A sequence \(\pi = (d_1, \dots, d_n)\) of nonnegative integers is graphic if there exists a graph \(G\) with \(n\) vertices for which \(d_1, \dots, d_n\) are the degrees of its vertices. \(G\) is referred to as a realization of \(\pi\). Let \(P\) be a graph property. A graphic sequence \(\pi\) is potentially \(P\)-graphic if there exists a realization of \(\pi\) with the graph property \(P\). Similarly, \(\pi\) is forcibly \(P\)-graphic if all realizations of \(\pi\) have the property \(P\). We characterize potentially Halin graph-graphic sequences, forcibly Halin graph-graphic sequences, and forcibly cograph-graphic sequences.