Subgraph connectivity numbers of a graph

Abstract

$\hat \delta , \hat \lambda , \hat \kappa$and ω denote the maximum values of the minimum degree, edge-connectivity, vertex-connectivity and clique size, respectively, of the subgraphs of the graph G. These subgraph connectivity numbers are shown to be ordered by ω-1 $\leqslant \hat \kappa \leqslant \hat \lambda \leqslant \hat \delta$for any graph. Extremal variations between these subgraph connectivity numbers of a graph are investigated. The constraint $\hat \delta \leqslant 2 \hat \lambda$− 1 is derived and equality is shown possible for certain graphs having 1 ≤ $\leqslant \hat \lambda \leqslant 6$. Extremal relations between the chromatic number and the subgraph connectivity numbers are investigated. Computationally, the cut lemmas are described and shown to yield polynomial bounded algorithms for computing $\hat \delta , \hat \lambda and \hat \kappa$.