Roman Bondage Number of a Graph

TL;DR

This paper introduces the Roman bondage number, a new graph invariant related to Roman domination, establishes bounds, calculates exact values for specific graphs, and proves the associated decision problem is NP-hard.

Contribution

It defines the Roman bondage number, provides bounds and exact values for certain graphs, and proves NP-hardness of the related decision problem.

Findings

Bounds for the Roman bondage number are established.

Exact values are determined for several classes of graphs.

Deciding the Roman bondage number is NP-hard even for bipartite graphs.

Abstract

The Roman dominating function on a graph $G=(V,E)$ is a function $f: V\rightarrow\{0,1,2\}$ such that each vertex $x$ with $f(x)=0$ is adjacent to at least one vertex $y$ with $f(y)=2$. The value $f(G)=\sum\limits_{u\in V(G)} f(u)$ is called the weight of $f$. The Roman domination number $\gamma_{\rm R}(G)$ is defined as the minimum weight of all Roman dominating functions. This paper defines the Roman bondage number $b_{\rm R}(G)$ of a nonempty graph $G=(V,E)$ to be the cardinality among all sets of edges $B\subseteq E$ for which $\gamma_{\rm R}(G-B)>\gamma_{\rm R}(G)$. Some bounds are obtained for $b_{\rm R}(G)$, and the exact values are determined for several classes of graphs. Moreover, the decision problem for $b_{\rm R}(G)$ is proved to be NP-hard even for bipartite graphs.