Abstract
An odd cycle transversal (oct, for short) in a graph is a set of vertices whose deletion will leave a graph without any odd cycles. The Odd Cycle Transversal (OCT) problem takes an undirected graph G and a non-negative integer k as input, and the objective is to test if G has an oct of size at most k. The directed counterpart of the problem, Directed Odd Cycle Transversal (DOCT), where the input is a digraph and k, is defined analogously. When parameterized by k, OCT is known to be FPT [Reed et al., Oper. Res. Lett., 2004] whereas DOCT was recently shown to be W[1]-hard [Lokshtanov et al., SODA, 2020].
A mixed graph is a graph that contains both directed and undirected edges. In this paper, we study the Mixed Odd Cycle Transversal (MOCT) problem, i.e., OCT on mixed graphs. And we show that MOCT admits a fixed-parameter tractable algorithm when parameterized by \(k+\ell \), where \(\ell \) is the number of directed edges in the input mixed graph. In the course of designing our algorithm for MOCT, we also design a fixed-parameter tractable algorithm for a variant of the well-known Multiway Cut problem, which might be of independent interest.
Lawqueen Kanesh is supported in part by NRF Fellowship for AI grant [R-252-000-B14-281] and by Defense Service Organization, Singapore. Saket Saurabh is supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement no. 819416), and Swarnajayanti Fellowship (no. DST/SJF/MSA01/2017-18). Jayakrishnan Madathil is supported by the Chennai Mathematical Institute and the Infosys Foundation. Part of this work was done while Jayakrishnan Madathil was at the Indian Institute of Technology Gandhinagar, supported by an IITGN-Early Career Fellowship. The authors would like to thank Komal Muluk for several helpful discussions on the MOCT problem, as well as for several suggestions that improved the presentation of this article.
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Das, A., Kanesh, L., Madathil, J., Saurabh, S. (2021). Odd Cycle Transversal in Mixed Graphs. In: Kowalik, Ł., Pilipczuk, M., Rzążewski, P. (eds) Graph-Theoretic Concepts in Computer Science. WG 2021. Lecture Notes in Computer Science(), vol 12911. Springer, Cham. https://doi.org/10.1007/978-3-030-86838-3_10
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