The bondage number of graphs on topological surfaces and Teschner's conjecture

The bondage number of a graph is the smallest number of its edges whose removal results in a graph having a larger domination number. We provide constant upper bounds for the bondage number of graphs on topological surfaces, improve upper bounds for the bondage number in terms of the maximum vertex degree and the orientable and non-orientable genera of the graph, and show tight lower bounds for the number of vertices of graphs 2-cell embeddable on topological surfaces of a given genus. Also, we provide stronger upper bounds for graphs with no triangles and graphs with the number of vertices larger than a certain threshold in terms of the graph genera. This settles Teschner's Conjecture in positive for almost all graphs.Comment: 21 pages; Original version from January 201

Upper bounds for the bondage number of graphs on topological surfaces

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 $\Delta(G)$ and embeddable on an orientable surface of genus h and a non-orientable surface of genus k, $b(G)\le \min\{\Delta(G)+h+2, \Delta(G)+k+1\}$. This generalizes known upper bounds for planar and toroidal graphs.Comment: 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, 201

On the Roman Bondage Number of Graphs on surfaces

A Roman dominating function on a graph $G$ is a labeling $f : V(G) \rightarrow \{0, 1, 2\}$ such that every vertex with label $0$ has a neighbor with label $2$. The Roman domination number, $\gamma_R(G)$, of $G$ is the minimum of $\Sigma_{v\in V (G)} f(v)$ over such functions. The Roman bondage number $b_R(G)$ is the cardinality of a smallest set of edges whose removal from $G$ results in a graph with Roman domination number not equal to $\gamma_R(G)$. In this paper we obtain upper bounds on $b_{R}(G)$ in terms of (a) the average degree and maximum degree, and (b) Euler characteristic, girth and maximum degree. We also show that the Roman bondage number of every graph which admits a $2$-cell embedding on a surface with non negative Euler characteristic does not exceed $15$.Comment: 5 page

Upper bounds for domination related parameters in graphs on surfaces

AbstractIn this paper we give tight upper bounds on the total domination number, the weakly connected domination number and the connected domination number of a graph in terms of order and Euler characteristic. We also present upper bounds for the restrained bondage number, the total restrained bondage number and the restricted edge connectivity of graphs in terms of the orientable/nonorientable genus and maximum degree

An improved upper bound for the bondage number of graphs on surfaces

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. Recently Gagarin and Zverovich showed that, for a graph $G$ with maximum degree $\Delta(G)$ and embeddable on an orientable surface of genus $h$ and a non-orientable surface of genus $k$, $b(G)\leq\min\{\Delta(G)+h+2,\Delta+k+1\}$. They also gave examples showing that adjustments of their proofs implicitly provide better results for larger values of $h$ and $k$. In this paper we establish an improved explicit upper bound for $b(G)$, using the Euler characteristic $\chi$ instead of the genera $h$ and $k$, with the relations $\chi=2-2h$ and $\chi=2-k$. We show that $b(G)\leq\Delta(G)+\lfloor r\rfloor$ for the case $\chi\leq0$ (i.e. $h\geq1$ or $k\geq2$), where $r$ is the largest real root of the cubic equation $z^3+2z^2+(6\chi-7)z+18\chi-24=0$. Our proof is based on the technique developed by Carlson-Develin and Gagarin-Zverovich, and includes some elementary calculus as a new ingredient. We also find an asymptotically equivalent result $b(G)\leq\Delta(G)+\lceil\sqrt{12-6\chi\,}-1/2\rceil$ for $\chi\leq0$, and a further improvement for graphs with large girth.Comment: 8 pages, to appear in Discrete Mathematic