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Double Roman domination in graphs: algorithmic complexity | ||
| Communications in Combinatorics and Optimization | ||
| مقاله 5، دوره 8، شماره 3، آذر 2023، صفحه 491-503 اصل مقاله (613.96 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2022.27661.1309 | ||
| نویسنده | ||
| Abolfazl Poureidi* | ||
| Shahrood University of Technology | ||
| چکیده | ||
| Let $G=(V,E)$ be a graph. A double Roman dominating function (DRDF) of $G $ is a function $f:V\to \{0,1,2,3\}$ such that, for each $v\in V$ with $f(v)=0$, there is a vertex $u $ adjacent to $v$ with $f(u)=3$ or there are vertices $x$ and $y $ adjacent to $v$ such that $f(x)=f(y)=2$ and for each $v\in V$ with $f(v)=1$, there is a vertex $u $ adjacent to $v$ with $f(u)>1$. The weight of a DRDF $f$ is $ f (V) =\sum_{ v\in V} f (v)$. Let $n$ and $k$ be integers such that $3\leq 2k+ 1 \leq n$. The generalized Petersen graph $GP (n, k)=(V,E) $ is the graph with $V=\{u_1, u_2,\ldots, u_n\}\cup\{v_1, v_2,\ldots, v_n\}$ and $E=\{u_iu_{i+1}, u_iv_i, v_iv_{i+k}: 1 \leq i \leq n\}$, where addition is taken modulo $n$. In this paper, we firstly prove that the decision problem associated with double Roman domination is NP-omplete even restricted to planar bipartite graphs with maximum degree at most 4. Next, we give a dynamic programming algorithm for computing a minimum DRDF (i.e., a DRDF with minimum weight along all DRDFs) of $GP(n,k )$ in $O(n81^k)$ time and space and so a minimum DRDF of $GP(n,O(1))$ can be computed in $O( n)$ time and space. | ||
| کلیدواژهها | ||
| Double Roman dominating function؛ Algorithm؛ Dynamic programming؛ Generalized Petersen graph | ||
| مراجع | ||
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