The Binding Number of Trees and \(K(1,3)\)-free Graphs

The binding number of a graph \(G\) is defined to be the minimum of \(|N(S)|/|S|\) taken over all nonempty \(S \subseteq V(G)\) such that \(N(S) \neq V(G)\). In this paper, another look is taken at the basic properties of the binding number. Several bounds are established, including ones linking the binding number of a tree to the “distribution” of its end-vertices. Further, it is established that under some simple conditions, \(K_{1,3}\)-free graphs have binding number equal to \((p(G) – 1)/(p(G) – \delta(G))\) and applications of this are considered.