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Algorithmic complexity of triple Roman dominating functions on graphs | ||
| Communications in Combinatorics and Optimization | ||
| مقاله 4، دوره 9، شماره 2، شهریور 2024، صفحه 217-232 اصل مقاله (809.44 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2023.27884.1385 | ||
| نویسندگان | ||
| Abolfazl Poureidi* ؛ Jafar Fathali | ||
| Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Iran | ||
| چکیده | ||
| Given a graph $G=(V,E)$, a function $f:V\to \{0,1,2,3,4\}$ is a triple Roman dominating function (TRDF) of $G$, for each vertex $v\in V$, (i) if $f (v ) = 0 $, then $v$ must have either one neighbour in $V_4$, or either two neighbours in $V_2 \cup V_3$ (one neighbour in $V_3$) or either three neighbours in $V_2 $, (ii) if $f (v ) = 1 $, then $v$ must have either one neighbour in $V_3 \cup V_4$ or either two neighbours in $V_2 $, and if $f (v ) = 2 $, then $v$ must have one neighbour in $V_2 \cup V_3\cup V_4$. The triple Roman domination number of $G$ is the minimum weight of an TRDF $f$ of $G$, where the weight of $f$ is $\sum_{v\in V}f(v)$. The triple Roman domination problem is to compute the triple Roman domination number of a given graph. In this paper, we study the triple Roman domination problem. We show that the problem is NP-complete for the star convex bipartite and the comb convex bipartite graphs and is APX-complete for graphs of degree at~most~4. We propose a linear-time algorithm for computing the triple Roman domination number of proper interval graphs. We also give an $( 2 H(\Delta(G)+1) -1 )$-approximation algorithm for solving the problem for any graph $G$, where $ \Delta(G)$ is the maximum degree of $G$ and $H(d)$ denotes the first $d$ terms of the harmonic series. In addition, we prove that for any $\varepsilon>0$ there is no $(1/4-\varepsilon)\ln|V|$-approximation polynomial-time algorithm for solving the problem on bipartite and split graphs, unless NP $\subseteq$ DTIME $(|V|^{O(\log\log|V |)})$. | ||
| کلیدواژهها | ||
| Triple Roman domination؛ Approximation algorithm؛ NP-complete؛ Proper interval graph؛ APX-complete | ||
| مراجع | ||
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