Maximal and Minimal Vertex-Critical Graphs of Diameter Two

Abstract

A graph is vertex-critical if deleting any vertex increases its diameter. We construct, for eachν≥5 exceptν=6, a vertex-critical graph of diameter two onνvertices with at least 12ν2-2ν3/2+c1νedges, wherec1is some constant. We show that such a graph contains at most 12ν2-(2/2)ν3/2+c2νedges, wherec2is some constant. We also construct, for eachν≥5 exceptν=6, a vertex-critical graph of diameter two onνvertices with at most 12 (5ν-12) edges. We show that such a graph must contain at least 12 (5ν-29) edges. © 1998 Academic Press.