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Total Roman domination and total domination in unit disk graphs | ||
| Communications in Combinatorics and Optimization | ||
| مقاله 6، دوره 10، شماره 4، اسفند 2025، صفحه 803-823 اصل مقاله (852.21 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2024.28647.1650 | ||
| نویسندگان | ||
| Sasmita Rout1؛ Pawan Kumar Mishra2؛ Gautam Kumar Das* 1 | ||
| 1Department of Mathematics, Indian Institute of Technology Guwahati, Assam, India | ||
| 2Department of Computer Science and Engineering, IIIT Guwahati, Assam, India | ||
| چکیده | ||
| Let $G=(V,E)$ be a simple, undirected and connected graph. A Roman dominating function (RDF) on the graph $G$ is a function $f:V\rightarrow\{0,1,2\}$ such that each vertex $v\in V$ with $f(v)=0$ is adjacent to at least one vertex $u\in V$ with $f(u)=2$. A total Roman dominating function (TRDF) of $G$ is a function $f:V\rightarrow\{0,1,2\}$ such that $(i)$ it is a Roman dominating function, and $(ii)$ the vertices with non-zero weights induce a subgraph with no isolated vertex. The total Roman dominating set (TRDS) problem is to minimize the associated weight, $f(V)=\sum_{u\in V} f(u)$, called the total Roman domination number ($\gamma_{tR}(G)$). Similarly, a subset $S\subseteq V$ is said to be a total dominating set (TDS) on the graph $G$ if $(i)$ $S$ is a dominating set of $G$, and $(ii)$ the induced subgraph $G[S]$ does not have any isolated vertex. The objective of the TDS problem is to minimize the cardinality of the TDS of a given graph. The TDS problem is NP-complete for general graphs. In this paper, we propose a simple $10.5\operatorname{-}$factor approximation algorithm for TRDS problem in UDGs. The running time of the proposed algorithm is $O(|V|\log k)$, where $k$ is the number of vertices with weights $2$. It is an improvement over the best-known $12$-factor approximation algorithm with running time $O(|V|\log k)$ available in the literature. Next, we propose another algorithm for the TDS problem in UDGs, which improves the previously best-known approximation factor from $8$ to $7.79$. The running time of the proposed algorithm is $O(|V|+|E|)$. | ||
| کلیدواژهها | ||
| Total Domination؛ Total Roman Domination؛ Approximation Algorithms؛ Unit Disk Graphs | ||
| مراجع | ||
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