Uniformly 3‐connected graphs

Abstract

Abstract Let be a positive integer. A graph is said to be uniformly ‐connected if between any pair of vertices the maximum number of independent paths is exactly . Dawes showed that all minimally 3‐connected graphs can be constructed from such that every graph in each intermediate step is also minimally 3‐connected. In this paper, we generalize Dawes' result to uniformly 3‐connected graphs. We give a constructive characterization of the class of uniformly 3‐connected graphs which differs from the characterization provided by Göring et al., where their characterization requires the set of all 3‐connected and 3‐regular graphs as a starting set, the new characterization requires only the graph . Eventually, we obtain a tight bound on the number of edges in uniformly 3‐connected graphs.