Total Roman bondage number of a graph

TL;DR

This paper introduces the total Roman bondage number of a graph, establishes its computational complexity, derives bounds, characterizes specific cases, and computes exact values for common graph classes.

Contribution

It defines the total Roman bondage number, proves NP-completeness of related decision problems, and provides bounds, characterizations, and exact values for various graph types.

Findings

Deciding whether $b_{tR}(G) \,\leq k$ is NP-complete.

Sharp bounds for $\,\gamma_{tR}(G)$ after edge removal.

Exact values for complete, bipartite, and special graphs.

Abstract

A total Roman dominating function (TRDF) on a graph $G$ with no isolated vertices is a function $f:V(G)\to\{0,1,2\}$ such that every vertex $v$ with $f(v)=0$ has a neighbor assigned $2$, and the subgraph induced by $\{v:f(v)>0\}$ has no isolated vertices. The total Roman domination number $\gamma_{tR}(G)$ is the minimum weight of a TRDF on $G$. The total Roman bondage number $b_{tR}(G)$ is the minimum cardinality of an edge set $E'\subseteq E(G)$ such that $G-E'$ has no isolated vertices and $\gamma_{tR}(G-E')>\gamma_{tR}(G)$; if no such $E'$ exists, $b_{tR}(G)=\infty$. We prove that deciding whether $b_{tR}(G)\leq k$ is NP-complete for arbitrary graphs. We establish sharp bounds, including $\gamma_{tR}(G)+1\leq \gamma_{tR}(G-B)\leq \gamma_{tR}(G)+2$ for any $b_{tR}(G)$-set $B$ (both sharp), and $b_{tR}(G)\geq \max\{\delta(G),b(G)\}$ when $\gamma_{tR}(G)=3\beta(G)$. We characterize graphs with $b_{tR}(G)=\infty$ and provide a necessary and sufficient condition for $b_{tR}(G)=1$. Exact values are determined for complete graphs, complete bipartite graphs, brooms, double brooms, wheels and wounded spiders. Further upper bounds are given in terms of order, diameter, girth, and structural features.