Domination in designs

TL;DR

This paper explores domination numbers in the incidence graphs of combinatorial designs, providing bounds, exact values for specific designs, and proposing conjectures about Steiner triple systems.

Contribution

It introduces new bounds and exact values for domination numbers in various combinatorial designs, especially Steiner systems, and proposes a conjecture on Steiner triple systems.

Findings

Finite projective planes have domination number 2n.

Symmetric (v,k,λ)-designs satisfy γ ≤ 2k.

Steiner triple systems have γ ≥ (2/3)v - 1.

Abstract

We commence the study of domination in the incidence graphs of combinatorial designs. Let $D$ be a combinatorial design and denote by $\gamma(D)$ the domination number of the incidence (Levy) graph of $D$. We obtain a number of results about the domination numbers of various kinds of designs. For instance, a finite projective plane of order $n$, which is a symmetric $(n^{2}+n+1,n+1,1)$-design, has $\gamma=2n$. %We also show that for any symmetric $(v,k,\lambda)$-design it holds that $\gamma \leq 2k$. We study at depth the domination numbers of Steiner systems and in particular of Steiner triple systems. We show that a $STS(v)$ has $\gamma \geq \frac{2}{3}v-1$ and also obtain a number of upper bounds. The tantalizing conjecture that all Steiner triple systems on $v$ vertices have the same domination number is proposed and is verified up to $v \leq 15$. The structure of minimal dominating sets is also investigated, both for its own sake and as a tool in deriving lower bounds on $\gamma$. Finally, a number of open questions are proposed.