A Linear Kernel for Planar Total Dominating Set
A total dominating set of a graph $G=(V,E)$ is a subset $D \subseteq V$ such that every vertex in $V$ is adjacent to some vertex in $D$. Finding a total dominating set of minimum size is NP-hard on planar graphs and W[2]-complete on general graphs when parameterized by the solution size. By the meta...
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| Vydáno v: | Discrete Mathematics and Theoretical Computer Science Ročník 20 no. 1; číslo Discrete Algorithms; s. 1 |
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| Hlavní autoři: | , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
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DMTCS
01.05.2018
Discrete Mathematics & Theoretical Computer Science |
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| ISSN: | 1365-8050, 1462-7264, 1365-8050 |
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| Abstract | A total dominating set of a graph $G=(V,E)$ is a subset $D \subseteq V$ such that every vertex in $V$ is adjacent to some vertex in $D$. Finding a total dominating set of minimum size is NP-hard on planar graphs and W[2]-complete on general graphs when parameterized by the solution size. By the meta-theorem of Bodlaender et al. [J. ACM, 2016], there exists a linear kernel for Total Dominating Set on graphs of bounded genus. Nevertheless, it is not clear how such a kernel can be effectively constructed, and how to obtain explicit reduction rules with reasonably small constants. Following the approach of Alber et al. [J. ACM, 2004], we provide an explicit kernel for Total Dominating Set on planar graphs with at most $410k$ vertices, where $k$ is the size of the solution. This result complements several known constructive linear kernels on planar graphs for other domination problems such as Dominating Set, Edge Dominating Set, Efficient Dominating Set, Connected Dominating Set, or Red-Blue Dominating Set.
Comment: 33 pages, 13 figures |
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| AbstractList | A total dominating set of a graph $G=(V,E)$ is a subset $D \subseteq V$ such that every vertex in $V$ is adjacent to some vertex in $D$. Finding a total dominating set of minimum size is NP-hard on planar graphs and W[2]-complete on general graphs when parameterized by the solution size. By the meta-theorem of Bodlaender et al. [J. ACM, 2016], there exists a linear kernel for Total Dominating Set on graphs of bounded genus. Nevertheless, it is not clear how such a kernel can be effectively constructed, and how to obtain explicit reduction rules with reasonably small constants. Following the approach of Alber et al. [J. ACM, 2004], we provide an explicit kernel for Total Dominating Set on planar graphs with at most $410k$ vertices, where $k$ is the size of the solution. This result complements several known constructive linear kernels on planar graphs for other domination problems such as Dominating Set, Edge Dominating Set, Efficient Dominating Set, Connected Dominating Set, or Red-Blue Dominating Set.
Comment: 33 pages, 13 figures A total dominating set of a graph G = (V, E) is a subset D [??] V such that every vertex in V is adjacent to some vertex in D. Finding a total dominating set of minimum size is NP-hard on planar graphs and W [2] -complete on general graphs when parameterized by the solution size. [B.sub.y] the meta-theorem of Bodlaender et al. [J. ACM, 2016], there exists a linear kernel for TOTAL DOMINATING SET on graphs of bounded genus. Nevertheless, it is not clear how such a kernel can be effectively constructed, and how to obtain explicit reduction rules with reasonably small constants. Following the approach of Alber et al. [J. ACM, 2004], we provide an explicit kernel for TOTAL DOMINATING SET on planar graphs with at most 410k vertices, where k is the size of the solution. This result complements several known constructive linear kernels on planar graphs for other domination problems such as DOMINATING SET, EDGE DOMINATING SET, EFFICIENT DOMINATING SET, CONNECTED DOMINATING SET, or RED-BLUE DOMINATING SET. Key words: parameterized complexity, planar graphs, linear kernels, total domination A total dominating set of a graph $G=(V,E)$ is a subset $D \subseteq V$ such that every vertex in $V$ is adjacent to some vertex in $D$. Finding a total dominating set of minimum size is NP-hard on planar graphs and W[2]-complete on general graphs when parameterized by the solution size. By the meta-theorem of Bodlaender et al. [J. ACM, 2016], there exists a linear kernel for Total Dominating Set on graphs of bounded genus. Nevertheless, it is not clear how such a kernel can be effectively constructed, and how to obtain explicit reduction rules with reasonably small constants. Following the approach of Alber et al. [J. ACM, 2004], we provide an explicit kernel for Total Dominating Set on planar graphs with at most $410k$ vertices, where $k$ is the size of the solution. This result complements several known constructive linear kernels on planar graphs for other domination problems such as Dominating Set, Edge Dominating Set, Efficient Dominating Set, Connected Dominating Set, or Red-Blue Dominating Set. A total dominating set of a graph G = (V, E) is a subset D [??] V such that every vertex in V is adjacent to some vertex in D. Finding a total dominating set of minimum size is NP-hard on planar graphs and W [2] -complete on general graphs when parameterized by the solution size. [B.sub.y] the meta-theorem of Bodlaender et al. [J. ACM, 2016], there exists a linear kernel for TOTAL DOMINATING SET on graphs of bounded genus. Nevertheless, it is not clear how such a kernel can be effectively constructed, and how to obtain explicit reduction rules with reasonably small constants. Following the approach of Alber et al. [J. ACM, 2004], we provide an explicit kernel for TOTAL DOMINATING SET on planar graphs with at most 410k vertices, where k is the size of the solution. This result complements several known constructive linear kernels on planar graphs for other domination problems such as DOMINATING SET, EDGE DOMINATING SET, EFFICIENT DOMINATING SET, CONNECTED DOMINATING SET, or RED-BLUE DOMINATING SET. |
| Audience | Academic |
| Author | Garnero, Valentin Sau, Ignasi |
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| Snippet | A total dominating set of a graph $G=(V,E)$ is a subset $D \subseteq V$ such that every vertex in $V$ is adjacent to some vertex in $D$. Finding a total... A total dominating set of a graph G = (V, E) is a subset D [??] V such that every vertex in V is adjacent to some vertex in D. Finding a total dominating set... |
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| SubjectTerms | 05c85, 05c10 computer science - data structures and algorithms Fuzzy sets g.2.2 Graph theory Kernel functions Mathematical research Mathematics Set theory |
| Title | A Linear Kernel for Planar Total Dominating Set |
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