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

Protecting a Graph with Mobile Guards

Mobile guards on the vertices of a graph are used to defend it against attacks on either its vertices or its edges. Various models for this problem have been proposed. In this survey we describe a number of these models with particular attention to the case when the attack sequence is infinitely long and the guards must induce some particular configuration before each attack, such as a dominating set or a vertex cover. Results from the literature concerning the number of guards needed to successfully defend a graph in each of these problems are surveyed.Comment: 29 pages, two figures, surve

Roman {2}-Bondage Number of a Graph

For a given graph G=(V, E), a Roman {2}-dominating function f : V (G) → {0, 1, 2} has the property that for every vertex u with f(u) = 0, either u is adjacent to a vertex assigned 2 under f, or is adjacent to at least two vertices assigned 1 under f. The Roman {2}-domination number of G, γ{R2}(G), is the minimum of Σu∈V (G)f(u) over all such functions. In this paper, we initiate the study of the problem of finding Roman {2}-bondage number of G. The Roman {2}-bondage number of G, b{R2}, is defined as the cardinality of a smallest edge set E′ ⊆ E for which γ{R2}(G − E′) > γ{R2}(G). We first demonstrate complexity status of the problem by proving that the problem is NP-Hard. Then, we derive useful parametric as well as fixed upper bounds on the Roman {2}-bondage number of G. Specifically, it is known that the Roman bondage number of every planar graph does not exceed 15 (see [S. Akbari, M. Khatirinejad and S. Qajar, A note on the Roman bondage number of planar graphs, Graphs Combin. 29 (2013) 327–331]). We show that same bound will be preserved while computing the Roman {2}-bondage number of such graphs. The paper is then concluded by computing exact value of the parameter for some classes of graphs

Roman {2}-Bondage Number of a Graph

For a given graph G=(V, E), a Roman {2}-dominating function f : V (G) → {0, 1, 2} has the property that for every vertex u with f(u) = 0, either u is adjacent to a vertex assigned 2 under f, or is adjacent to at least two vertices assigned 1 under f. The Roman {2}-domination number of G, γ{R2}(G), is the minimum of Σu∈V (G) f(u) over all such functions. In this paper, we initiate the study of the problem of finding Roman {2}-bondage number of G. The Roman {2}-bondage number of G, b{R2}, is defined as the cardinality of a smallest edge set E′ ⊆ E for which γ{R2}(G − E′) > γ{R2}(G). We first demonstrate complexity status of the problem by proving that the problem is NP-Hard. Then, we derive useful parametric as well as fixed upper bounds on the Roman {2}-bondage number of G. Specifically, it is known that the Roman bondage number of every planar graph does not exceed 15 (see [S. Akbari, M. Khatirinejad and S. Qajar, A note on the Roman bondage number of planar graphs, Graphs Combin. 29 (2013) 327–331]). We show that same bound will be preserved while computing the Roman {2}-bondage number of such graphs. The paper is then concluded by computing exact value of the parameter for some classes of graphs