Complexity of Paired Domination Problems on Circle and $k$-Polygon Graphs
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866915166008901632 |
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| author | Mu, Ta-Yu Lin, Ching-Chi |
| author_facet | Mu, Ta-Yu Lin, Ching-Chi |
| contents | A set $D \subseteq V$ is a dominating set of a graph $G$ if every vertex in $V - D$ is adjacent to at least one vertex in $D$. A dominating set $D$ is a paired-dominating set if the subgraph of $G$ induced by $D$ contains a perfect matching. In this paper, we prove that determining the minimum paired-dominating set in circle graphs is NP-complete. We further present an $O(n(\frac{n}{k^2-k})^{2k^2-2k})$-time algorithm for finding the minimum paired-dominating set in $k$-polygon graphs, a subclass of circle graphs. Additionally, we refine the existing algorithm of Elmallah and Stewart for computing the minimum dominating set in $k$-polygon graphs, reducing its time complexity from $O(n^{4k^2+3})$ to $O(n^{3k-5})$, and further extend it to find the minimum total dominating set. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_19473 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Complexity of Paired Domination Problems on Circle and $k$-Polygon Graphs Mu, Ta-Yu Lin, Ching-Chi Data Structures and Algorithms Computational Complexity Combinatorics A set $D \subseteq V$ is a dominating set of a graph $G$ if every vertex in $V - D$ is adjacent to at least one vertex in $D$. A dominating set $D$ is a paired-dominating set if the subgraph of $G$ induced by $D$ contains a perfect matching. In this paper, we prove that determining the minimum paired-dominating set in circle graphs is NP-complete. We further present an $O(n(\frac{n}{k^2-k})^{2k^2-2k})$-time algorithm for finding the minimum paired-dominating set in $k$-polygon graphs, a subclass of circle graphs. Additionally, we refine the existing algorithm of Elmallah and Stewart for computing the minimum dominating set in $k$-polygon graphs, reducing its time complexity from $O(n^{4k^2+3})$ to $O(n^{3k-5})$, and further extend it to find the minimum total dominating set. |
| title | Complexity of Paired Domination Problems on Circle and $k$-Polygon Graphs |
| topic | Data Structures and Algorithms Computational Complexity Combinatorics |
| url | https://arxiv.org/abs/2411.19473 |