On the Hull Number of Triangle-Free Graphs

A set of vertices C in a graph is convex if it contains all vertices which lie on shortest paths between vertices in C. The convex hull of a set of vertices S is the smallest convex set containing S. The hull number h(G) of a graph G is the smallest cardinality of a set of vertices whose convex hull is the vertex set of G. For a connected triangle-free graph G of order n and diameter d ≥ 3, we prove that h(G) ≤ (n− d+ 3)/3, if G has minimum degree at least 3 and that h(G) ≤ 2(n− d+ 5)/7, if G is cubic. Furthermore, for a connected graph G of order n, girth g ≥ 4, minimum degree at least 2, and diameter d, we prove h(G) ≤ 2 + (n− d− 1)/ ⌈ g−1 2 ⌉ . All bounds are best possible.