The price of connectivity for cycle transversals

Hartinger, Tatiana Romina; Johnson, Matthew; Milanič, Martin; Paulusma, Daniël

doi:10.1016/j.ejc.2016.06.003

Cited by 13 publications

(15 citation statements)

References 24 publications

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“…We now consider the Odd Cycle Transversal problem and its connected variant. Again we can use a result on the price of connectivity, which is also covered by Lemma 10 of [18]. Proof.…”

Section: Odd Cycle Transversal and Connected Odd Cycle Transversalmentioning

confidence: 99%

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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“…We now consider the Odd Cycle Transversal problem and its connected variant. Again we can use a result on the price of connectivity, which is also covered by Lemma 10 of [18]. Proof.…”

Section: Odd Cycle Transversal and Connected Odd Cycle Transversalmentioning

confidence: 99%

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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“…The complexity of Feedback Vertex Set and Connected Feedback Vertex Set is unknown when restricted to P r -free graphs for r ≥ 5. For every s ≥ 1, both problems and their connected variants are polynomial-time solvable on sP 2 -free graphs [10], using the price of connectivity for feedback vertex set [2,21]. 5…”

Section: Introductionmentioning

confidence: 99%

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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“…To the best of the authors' knowledge, this work represents the first systematic study of the problem of classifying hereditary graph classes with respect to the existence of a polynomial bound on the number of minimal separators of the graphs in the class. Dichotomy studies for many other problems in mathematics and computer science are available in the literature in general, as well as within the field of graph theory, for properties such as boundedness of the clique-width [7][8][9]12], price of connectivity [20], and polynomial-time solvability of various algorithmic problems such as Chromatic Number [17,29,31], Graph Homomorphism [22], Graph Isomorphism [42], and Dominating Set [32].…”

Section: Introductionmentioning

confidence: 99%

Minimal Separators in Graph Classes Defined by Small Forbidden Induced Subgraphs

Milanič

Pivač

2019

Graph-Theoretic Concepts in Computer Science

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Minimal separators in graphs are an important concept in algorithmic graph theory. In particular, many problems that are NP-hard for general graphs are known to become polynomial-time solvable for classes of graphs with a polynomially bounded number of minimal separators. Several well-known graph classes have this property, including chordal graphs, permutation graphs, circular-arc graphs, and circle graphs. We perform a systematic study of the question which classes of graphs defined by small forbidden induced subgraphs have a polynomially bounded number of minimal separators. We focus on sets of forbidden induced subgraphs with at most four vertices and obtain an almost complete dichotomy, leaving open only two cases.

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The price of connectivity for cycle transversals

Cited by 13 publications

References 24 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

Minimal Separators in Graph Classes Defined by Small Forbidden Induced Subgraphs

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