Total Roman bondage number of a graph
Fahimeh Khosh-Ahang Ghasr, Sakineh Nazari-Moghaddam
TL;DR
The paper studies the total Roman bondage number $b_{tR}(G)$, the minimum edge-deletion size that preserves connectivity while forcing an increase in the total Roman domination number $\gamma_{tR}(G)$. It establishes that deciding whether $b_{tR}(G)\le k$ is NP-hard via a 3-SAT reduction, and develops sharp bounds relating $\gamma_{tR}$ before and after edge deletions. It also characterizes graphs with $b_{tR}(G)=\infty$ or $b_{tR}(G)=1$, and determines exact values for several standard graph families such as complete graphs, complete bipartite graphs, wheels, brooms, and spiders. Additionally, the work derives further upper bounds in terms of order, diameter, girth, and structural features, deepening the understanding of robustness of total Roman domination under edge removals.
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 $γ_{tR}(G)$ 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 $γ_{tR}(G-E')>γ_{tR}(G)$; 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 $γ_{tR}(G)+1\leq γ_{tR}(G-B)\leq γ_{tR}(G)+2$ for any $b_{tR}(G)$-set $B$ (both sharp), and $b_{tR}(G)\geq \max\{δ(G),b(G)\}$ when $γ_{tR}(G)=3β(G)$. 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.