Total Roman bondage number of a graph
Year: 2026
Type: article
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.
Authors: F. Khosh-Ahang Ghasr, S. Nazari-Moghaddam
Institution: Ilam University
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $1,226
OpenAlex ID: W7128531643