Total Roman bondage number of a graph

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.

💡 Research Summary

The paper introduces and studies the total Roman bondage number (b_{tR}(G)) of a graph (G). A total Roman dominating function (TRDF) is a mapping (f:V(G)\rightarrow{0,1,2}) such that every vertex assigned 0 has a neighbor assigned 2, and the subgraph induced by the vertices with positive values contains no isolated vertex. The weight of a TRDF is (\omega(f)=|V_1|+2|V_2|) where (V_i={v\mid f(v)=i}); the minimum possible weight is the total Roman domination number (\gamma_{tR}(G)).

The total Roman bondage number (b_{tR}(G)) is defined as the smallest cardinality of an edge set (E’\subseteq E(G)) whose removal leaves a graph without isolated vertices and strictly increases (\gamma_{tR}). If no such set exists, (b_{tR}(G)=\infty).

Main contributions

1. Complexity – By a polynomial‑time reduction from 3‑SAT, the decision problem “Is (b_{tR}(G)\le k)?” is shown to be NP‑complete for arbitrary graphs. The construction uses a gadget (H_i) for each variable and a clause vertex (c_j) connected to the appropriate variable gadgets. It is proved that (\gamma_{tR}(G)=4n+3) (where (n) is the number of variables) iff the formula is satisfiable, and that in this case a single edge removal already raises (\gamma_{tR}), i.e., (b_{tR}(G)=1).

2. General bounds – For any minimum bondage set (B) we have
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