On the Roman bondage number of a graph

TL;DR

This paper investigates the Roman bondage number of graphs, proving its decision problem is NP-hard for bipartite graphs and establishing bounds and characterizations for this graph invariant.

Contribution

It introduces the NP-hardness of computing the Roman bondage number for bipartite graphs and provides bounds and characterizations for this parameter.

Findings

Decision problem for Roman bondage number is NP-hard for bipartite graphs.

Established sharp bounds for the Roman bondage number.

Characterized graphs that attain these bounds.

Abstract

A Roman dominating function on a graph $G=(V,E)$ is a function $f:V\rightarrow\{0,1,2\}$ such that every vertex $v\in V$ with $f(v)=0$ has at least one neighbor $u\in V$ with $f(u)=2$. The weight of a Roman dominating function is the value $f(V(G))=\sum_{u\in V(G)}f(u)$. The minimum weight of a Roman dominating function on a graph $G$ is called the Roman domination number, denoted by $\gamma_{R}(G)$. The Roman bondage number $b_{R}(G)$ of a graph $G$ with maximum degree at least two is the minimum cardinality of all sets $E'\subseteq E(G)$ for which $\gamma_{R}(G-E')>\gamma_R(G)$. In this paper, we first show that the decision problem for determining $b_{\rm R}(G)$ is NP-hard even for bipartite graphs and then we establish some sharp bounds for $b_{\rm R}(G)$ and characterizes all graphs attaining some of these bounds.