On the Roman Bondage Number of Graphs on surfaces

Abstract

A Roman dominating function on a graph $G$ is a labeling $f:V(G)\to \{0,1,2\}$ 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 $15$.

5 pages