Generating and Enumerating Digitally Convex Sets of Trees

A set S of vertices in a graph G with vertex set V is digitally convex if for every vertex \(v \in V\), \(Nv] \subseteq NS]\) implies \(v \in S\). We show that a vertex belongs to at most half of the digitally convex sets of a graph. Moreover, a vertex belongs to exactly half of the digitally convex sets if and only if it is simplicial. An algorithm that generates all digitally convex sets of a tree is described and sharp upper and lower bounds for the number of digitally convex sets of a tree are obtained. A closed formula for the number of digitally convex sets of a path is derived. It is shown how a binary cotree of a cograph can be used to enumerate its digitally convex sets in linear time.