PBIB-Designs Arising From BI-Connected Dominating Sets of Cubic Graphs of Order at Most 12

A (v, b, γbc, r, λᵢ)-design over regular graph $G = (V, E)$ of degree k is an ordered pair $D = (V, B)$, where $|V| = v$ and B is the set of bi-connected dominating sets of G called blocks such that two vertices α and β which are iᵗʰ associates occur together in λᵢ blocks, the numbers λᵢ being independent of the choice of the pair α and β. In this paper, we obtain Partially Balanced Incomplete Block (PBIB)-designs arising from bi-connected dominating sets in cubic graphs. Also, we give a complete list of PBIB-designs with respect to the bi-connected dominating sets for cubic graphs of order at most $12$. The discussion of non-existence of some designs corresponding to bi-connected dominating sets from certain graphs concludes the article.