Generating and Enumerating Digitally Convex Sets of Trees

Suppose V is a finite set and C a collection of subsets of V that contains ∅ and V and is closed under taking intersections. Then C is called a convexity and the ordered pair (V, C) is called an aligned space and the elements of C are referred to as convex sets. For a set S ⊆ V , the convex hull of S relative to C, denoted by CHC(S), is the smallest convex set containing S. A set S of vertices in a graph G with vertex set V is digitally convex if for every vertex v ∈ V , N [v] ⊆ N [S] implies v ∈ S. 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 established. It is shown how a binary cotree of a cograph can be used to enumerate its digitally convex sets in linear time.