On domination numbers of graph bundles
Blaž Zmazek and Janez Žerovnik
Journal of Applied Mathematics and Computing (JAMC) Volume 22, Number 1-2, pp. 39-48, 2006. ISSN 1598 - 5865

Abstract

Let $\gamma (G)$ be the domination number of a graph $G$. It is shown that for any $k\ge 0$ there exists a Cartesian graph bundle $B\bun F$ such that $\gamma (B\bun F)= \gamma(B) \gamma (F)-2k$. The domination numbers of Cartesian bundles of two cycles are determined exactly when the fibre graph is a triangle or a square. A statement similar to Vizing's conjecture on strong graph bundles is shown not to be true by proving the inequality $\dom (B\buns F)\le \dom (B) \dom (F)$ for strong graph bundles. Examples of graphs $B$ and $F$ with $\dom (B\buns F)< \dom (B) \dom (F)$ are given.

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