PBIB-designs and association schemes arising from minimum bi-connected dominating sets of some special classes of graphs

Abstract

A dominating set D of a connected graph $$G = (V, E)$$ is said to be bi-connected dominating set if the induced subgraphs of both $$\langle D \rangle $$ and $$\langle V-D \rangle $$ are connected. The bi-connected domination number $$\gamma _{bc}(G)$$ is the minimum cardinality of a bi-connected dominating set. A $$\gamma _{bc}$$ -set is a minimum bi-connected dominating set of G. In this paper, we obtain the Partially Balanced Incomplete Block (PBIB)-designs with m = 1, 2, 3, 4 and $$\lfloor \frac{p}{2}\rfloor $$ association schemes arising from $$\gamma _{bc}$$ -sets of some special classes of graphs.