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Vertex-edge Roman domination in graphs: complexity and algorithms | ||
| Communications in Combinatorics and Optimization | ||
| مقاله 3، دوره 8، شماره 1، خرداد 2023، صفحه 23-37 اصل مقاله (479.67 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2021.27163.1206 | ||
| نویسندگان | ||
| Manjay Kumar* ؛ Venkata Subba Reddy P | ||
| Department of Computer Science and Engineering, National Institute of Technology, Warangal, India | ||
| چکیده | ||
| For a simple, undirected graph $G(V,E)$, a function $h : V(G) \rightarrow \lbrace 0, 1, 2\rbrace$ such that each edge $ (u,v)$ of $G$ is either incident with a vertex with weight at least one or there exists a vertex $w$ such that either $(u,w) \in E(G)$ or $(v,w) \in E(G)$ and $h(w) = 2$, is called a vertex-edge Roman dominating function (ve-RDF) of $G$. For a graph $G$, the smallest possible weight of a ve-RDF of $G$ which is denoted by $\gamma_{veR}(G)$, is known as the \textit{vertex-edge Roman domination number} of $G$. The problem of determining $\gamma_{veR}(G)$ of a graph $G$ is called minimum vertex-edge Roman domination problem (MVERDP). In this article, we show that the problem of deciding if $G$ has a ve-RDF of weight at most $l$ for star convex bipartite graphs, comb convex bipartite graphs, chordal graphs and planar graphs is NP-complete. On the positive side, we show that MVERDP is linear time solvable for threshold graphs, chain graphs and bounded tree-width graphs. On the approximation point of view, a 2-approximation algorithm for MVERDP is presented. It is also shown that vertex cover and vertex-edge Roman domination problems are not equivalent in computational complexity aspects. Finally, an integer linear programming formulation for MVERDP is presented. | ||
| کلیدواژهها | ||
| Vertex-edge Roman-domination؛ Graph classes؛ NP-complete؛ Vertex cover؛ Integer linear programming | ||
| مراجع | ||
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