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- W4300972031 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." @default.
- W4300972031 created "2022-10-04" @default.
- W4300972031 creator A5001903817 @default.
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- W4300972031 date "2014-05-14" @default.
- W4300972031 modified "2023-09-27" @default.
- W4300972031 title "Domination in designs" @default.
- W4300972031 doi "https://doi.org/10.48550/arxiv.1405.3436" @default.
- W4300972031 hasPublicationYear "2014" @default.
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