Upper bounds for the bondage number of graphs on topological surfaces
Andrei Gagarin, Vadim Zverovich
备注：
10 pages; Updated version (April 2011); Presented at the 7th ECCC, Wolfville (Nova Scotia, Canada), May 4-6, 2011, and the 23rd BCC, Exeter (England, UK), July 3-8, 2011
The bondage number b(G) of a graph G is the smallest number of edges of G whose removal from G results in a graph having the domination number larger than that of G. We show that, for a graph G having the maximum vertex degree $Δ(G)$ and embeddable on an orientable surface of genus h and a non-orientable surface of genus k, $b(G)\le \min\{Δ(G)+h+2, Δ(G)+k+1\}$. This generalizes known upper bounds for planar and toroidal graphs.
The bondage number $b(G)$ of a graph $G$ is the smallest number of edges whose removal from $G$ results in a graph with larger domination number. Let $G$ be embeddable on a surface whose Euler characteristic $χ$ is as large as possible, and assume $χ\leq0$. Gagarin-Zverovich and Huang have recently found upper bounds of $b(G)$ in terms of the maximum degree $Δ(G)$ and the Euler characteristic $χ(G)=χ$. In this paper we prove a better upper bound $b(G)\leqΔ(G)+\lfloor t\rfloor$ where $t$ is the largest real root of the cubic equation $z^3 + z^2 + (3χ- 8)z + 9χ- 12=0$; this upper bound is asympt…
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