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| Main Authors: | , , |
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| Format: | Preprint |
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2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2206.07358 |
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| _version_ | 1866918239307563008 |
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| author | Krithika, R. Sharma, Roohani Tale, Prafullkumar |
| author_facet | Krithika, R. Sharma, Roohani Tale, Prafullkumar |
| contents | For a positive integer $\ell \geq 3$, the $C_\ell$-Contractibility problem takes as input an undirected simple graph $G$ and determines whether $G$ can be transformed into a graph isomorphic to $C_\ell$ (the induced cycle on $\ell$ vertices) using only edge contractions. Brouwer and Veldman [JGT 1987] showed that $C_4$-Contractibility is NP-complete in general graphs. It is easy to verify that $C_3$-Contractibility is polynomial-time solvable. Dabrowski and Paulusma [IPL 2017] showed that $C_{\ell}$-Contractibility is \NP-complete\ on bipartite graphs for $\ell = 6$ and posed as open problems the status of the problem when $\ell$ is 4 or 5. In this paper, we show that both $C_5$-Contractibility and $C_4$-Contractibility are NP-complete on bipartite graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_07358 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The Complexity of Contracting Bipartite Graphs into Small Cycles Krithika, R. Sharma, Roohani Tale, Prafullkumar Computational Complexity For a positive integer $\ell \geq 3$, the $C_\ell$-Contractibility problem takes as input an undirected simple graph $G$ and determines whether $G$ can be transformed into a graph isomorphic to $C_\ell$ (the induced cycle on $\ell$ vertices) using only edge contractions. Brouwer and Veldman [JGT 1987] showed that $C_4$-Contractibility is NP-complete in general graphs. It is easy to verify that $C_3$-Contractibility is polynomial-time solvable. Dabrowski and Paulusma [IPL 2017] showed that $C_{\ell}$-Contractibility is \NP-complete\ on bipartite graphs for $\ell = 6$ and posed as open problems the status of the problem when $\ell$ is 4 or 5. In this paper, we show that both $C_5$-Contractibility and $C_4$-Contractibility are NP-complete on bipartite graphs. |
| title | The Complexity of Contracting Bipartite Graphs into Small Cycles |
| topic | Computational Complexity |
| url | https://arxiv.org/abs/2206.07358 |