Domination number in block designs

TL;DR

This paper investigates the domination number of incidence graphs of combinatorial designs, proving and disproving several conjectures, and provides new insights into the structure of these graphs.

Contribution

It proves a conjecture and disproves another regarding domination numbers in incidence graphs, and confirms a third conjecture for block-transitive symmetric designs.

Findings

Proved a conjecture on domination number in incidence graphs.

Disproved a recent conjecture related to the same topic.

Validated a conjecture for block-transitive symmetric designs.

Abstract

Let $G=(V,E)$ be a simple connected graph. A set of vertices $S\subseteq V$ is said to be a dominating set if for any vertex in $V\setminus S$ is adjacent to at least one vertex in $S$. The domination number $\gamma(G)$ of $G$ is the minimum cardinality among all such sets. In this paper, we obtain some results on the domination number of the incidence graphs of combinatorial designs. In particular, we prove a conjecture and disprove another conjecture in a recent paper by Goldberg, Rajendraprasad and Mathew. We also prove a third conjecture by the same authors for block-transitive symmetric designs.