The Complexity of Contracting Bipartite Graphs into Small Cycles

R. Krithika; Roohani Sharma; Prafullkumar Tale

doi:10.46298/dmtcs.14658

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arXiv · 2206.07358

The Complexity of Contracting Bipartite Graphs into Small Cycles

R. Krithika·Roohani Sharma·Prafullkumar Tale

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Abstract

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.

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R. Krithika, Roohani Sharma, Prafullkumar Tale. 2022-06-15. The Complexity of Contracting Bipartite Graphs into Small Cycles. https://doi.org/10.46298/dmtcs.14658

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