Boundary classes for graph problems involving non-local properties

Munaro, Andrea

doi:10.1016/j.tcs.2017.06.012

Cited by 17 publications

(18 citation statements)

References 48 publications

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“…The above hardness results imply that Connected Feedback Vertex Set and Connected Odd Cycle Transversal are NP-complete on H-free graphs whenever H is a graph that contains a cycle or a claw. Due to the results of Munaro [23] the same holds for Connected Vertex Cover. Moreover, the same holds for the original variants Feedback Vertex Set and Odd Cycle Transversal but not for Vertex Cover, which is polynomial-time solvable for claw-free graphs.…”

Section: Our Resultsmentioning

confidence: 64%

“…Wanatabe, Kajita, and Onaga [32] proved NPcompleteness of Connected Vertex Cover for 3-connected graphs. More recently, Munaro [23] established NP-completeness of Connected Vertex Cover for line graphs of planar cubic bipartite graphs and for planar bipartite graphs of arbitrarily large girth. Ueno, Kajitani, and Gotoh [31] proved that Connected Vertex Cover can be solved in polynomial time for graphs of maximum degree at most 3 and for trees.…”

Section: Related Workmentioning

confidence: 99%

“…Proof. We reduce from Connected Vertex Cover restricted to graphs of girth at least p. Recall that Munaro [23] proved that this problem is NP-complete. Let G be a graph of girth at least p. We may assume without loss of generality that G has at least two edges.…”

Section: Lemmamentioning

confidence: 99%

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

Chiarelli

Hartinger

Johnson

et al. 2018

Theoretical Computer Science

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Abstract. 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.

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Section: Our Resultsmentioning

confidence: 64%

Section: Related Workmentioning

confidence: 99%

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

Chiarelli

Hartinger

Johnson

et al. 2018

Theoretical Computer Science

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“…[3,26]). We also note that Munro [28] showed that Feedback Vertex Set is NP-complete even on line graphs of planar cubic bipartite graphs. Unreferenced results directly follow from other results in the Remark 1.…”

Section: Connected Vertex Cover Extensionmentioning

confidence: 77%

On Cycle Transversals and Their Connected Variants in the Absence of a Small Linear Forest

et al. 2020

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A graph is H-free if it contains no induced subgraph isomorphic to H. We prove new complexity results for the two classical cycle transversal problems Feedback Vertex Set and Odd Cycle Transversal by showing that they can be solved in polynomial time on (sP1 + P3)-free graphs for every integer s ≥ 1. We show the same result for the variants Connected Feedback Vertex Set and Connected Odd Cycle Transversal. We also prove that the latter two problems are polynomial-time solvable on cographs; this was already known for Feedback Vertex Set and Odd Cycle Transversal. We complement these results by proving that Odd Cycle Transversal and Connected Odd Cycle Transversal are NP-complete on (P2 +P5, P6)-free graphs.called Connected Vertex Cover, Connected Feedback Vertex Set and Connected Odd Cycle Transversal, respectively. Garey and Johnson [15] proved that Connected Vertex Cover is NP-complete even on planar graphs of maximum degree 4 (see, for example, [14,31,36] for NPcompleteness results for other graph classes). Grigoriev and Sitters [18] proved that Connected Feedback Vertex Set is NP-complete even on planar graphs with maximum degree 9. More recently, Chiarelli et al. [10] proved that Connected Odd Cycle Transversal is NP-complete even on graphs of arbitrarily large girth and on line graphs.As all three decision problems and their connected variants are NP-complete, we can consider how to restrict the input to some special graph class in order to achieve tractability. Note that this approach is in line with the aforementioned results in the literature, where NP-completeness was proven on special graph classes. It is also in line with with, for instance, polynomial-time results for Connected Vertex Cover by Escoffier, Gourvès and Monnot [12] (for chordal graphs) and Ueno, Kajitani and Gotoh [35] (for graphs of maximum degree at most 3 and trees).Just as in most of these papers, we consider hereditary graph classes, that is, graph classes closed under vertex deletion. Hereditary graph classes form a rich framework that captures many well-studied graph classes. It is not difficult to see that every hereditary graph class G can be characterized by a (possibly infinite) set F G of forbidden induced subgraphs. If |F G | = 1, say F = {H}, then G is said to be monogenic, and every graph G ∈ G is said to be H-free. Considering monogenic graph classes can be seen as a natural first step for increasing our knowledge of the complexity of an NP-complete problem in a systematic way. Hence, we consider the following research question:How does the structure of a graph H influence the computational complexity of a graph transversal problem for input graphs that are H-free?Note that different graph transversal problems may behave differently on some class of H-free graphs. However, the general strategy for obtaining complexity results is to first try to prove that the restriction to H-free graphs is NP-complete whenever H contains a cycle or the claw (the 4-vertex star). This is usually done by showing, respectively, that the probl...

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“…Wanatabe et al [22] proved that Connected Vertex Cover is NP-complete even for 3-connected graphs. Very recently, Munaro [16] proved the same for line graphs of planar cubic bipartite graphs and for planar bipartite graphs of arbitrarily large girth, and Li et al [13] showed NP-completeness for 4-regular graphs. We now turn to tractable cases.…”

Section: Connected Vertex Covermentioning

confidence: 88%

Connected Vertex Cover for $$(sP_1+P_5)$$-Free Graphs

Johnson

Paesani

Paulusma

2018

Graph-Theoretic Concepts in Computer Science

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The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details.Abstract. The Connected Vertex Cover problem is to decide if a graph G has a vertex cover of size at most k that induces a connected subgraph of G. This is a well-studied problem, known to be NP-complete for restricted graph classes, and, in particular, for H-free graphs if H is not a linear forest. On the other hand, the problem is known to be polynomial-time solvable for sP2-free graphs for any integer s ≥ 1. We prove that it is also polynomial-time solvable for (sP1 + P5)-free graphs for every integer s ≥ 0.

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Boundary classes for graph problems involving non-local properties

Cited by 17 publications

References 48 publications

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

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

On Cycle Transversals and Their Connected Variants in the Absence of a Small Linear Forest

Connected Vertex Cover for $$(sP_1+P_5)$$-Free Graphs

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