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Algorithmic Aspects of Quasi-Total Roman Domination in Graphs | ||
| Communications in Combinatorics and Optimization | ||
| مقاله 10، دوره 7، شماره 1، شهریور 2022، صفحه 93-104 اصل مقاله (432.4 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2021.27126.1200 | ||
| نویسندگان | ||
| Venkata Subba Reddy P* 1؛ Mangal Vikas2 | ||
| 1National Institute of Technology Warangal | ||
| 2Department of Computer Science and Engineering, National Institute of Technology Warangal, India. | ||
| چکیده | ||
| For a simple, undirected, connected graph $G$($V,E$), a function $f : V(G) \rightarrow \lbrace 0, 1, 2 \rbrace$ which satisfies the following conditions is called a quasi-total Roman dominating function (QTRDF) of $G$ with weight $f(V(G))=\sum_{v \in V(G)} f(v)$. C1). Every vertex $u \in V(G)$ for which $f(u) = 0$ must be adjacent to at least one vertex $v$ with $f(v) = 2$, and C2). Every vertex $u \in V(G)$ for which $f(u) = 2$ must be adjacent to at least one vertex $v$ with $f(v) \geq 1$. For a graph $G$, the smallest possible weight of a QTRDF of $G$ denoted $\gamma_{qtR}(G)$ is known as the \textit{quasi-total Roman domination number} of $G$. The problem of determining $\gamma_{qtR}(G)$ of a graph $G$ is called minimum quasi-total Roman domination problem (MQTRDP). In this paper, we show that the problem of determining whether $G$ has a QTRDF of weight at most $l$ is NP-complete for split graphs, star convex bipartite graphs, comb convex bipartite graphs and planar graphs. On the positive side, we show that MQTRDP for threshold graphs, chain graphs and bounded treewidth graphs is linear time solvable. Finally, an integer linear programming formulation for MQTRDP is presented. | ||
| کلیدواژه‌ها | ||
| Domination number؛ quasi-total Roman domination؛ complexity classes؛ graph classes؛ linear programming | ||
| مراجع | ||
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