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. Keywords. Convex hull; convex set; geodetic number; graph; hull number

• Type: Journal
• Year: 2010
• Where: SIAMDM
• Authors: Mitre Costa Dourado, Fábio Protti, Dieter Rautenbach, Jayme Luiz Szwarcfiter
• Keywords: Connected Triangle-free Graph, Convex Hull, Hull Number, SIAMDM 2010