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Minimum connected transversals in graphs : new hardness results and tractable cases using the price of connectivity.

Chiarelli, Nina and Hartinger, Tatiana R. and Johnson, Matthew and Milanič, Martin and Paulusma, Daniël (2018) 'Minimum connected transversals in graphs : new hardness results and tractable cases using the price of connectivity.', Theoretical computer science., 705 . pp. 75-83.


We perform a systematic study in the computational complexity of the connected variant of three related transversal problems: Vertex Cover, Feedback Vertex Set, and Odd Cycle Transversal. Just like their original counterparts, these variants are NP-complete for general graphs. A graph G is H-free for some graph H if G contains no induced subgraph isomorphic to H. It is known that Connected Vertex Cover is NP-complete even for H-free graphs if H contains a claw or a cycle. We show that the two other connected variants also remain NP-complete if H contains a cycle or claw. In the remaining case H is a linear forest. We show that Connected Vertex Cover, Connected Feedback Vertex Set, and Connected Odd Cycle Transversal are polynomial-time solvable for sP2-free graphs for every constant s≥1. For proving these results we use known results on the price of connectivity for vertex cover, feedback vertex set, and odd cycle transversal. This is the first application of the price of connectivity that results in polynomial-time algorithms.

Item Type:Article
Full text:(AM) Accepted Manuscript
Available under License - Creative Commons Attribution Non-commercial No Derivatives.
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Publisher statement:© 2017 This manuscript version is made available under the CC-BY-NC-ND 4.0 license
Date accepted:27 September 2017
Date deposited:18 October 2017
Date of first online publication:04 October 2017
Date first made open access:04 October 2018

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