On line graphs of subcubic triangle-free graphs

Munaro, Andrea

doi:10.1016/j.disc.2017.01.006

Cited by 21 publications

(16 citation statements)

References 34 publications

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“…The latter result was recently extended to (sP 1 + P 5 )-free graphs for every s ≥ 0 [11,arXiv version]. We write H ⊆ i G to denote that H is an induced subgraph of G. We can now summarize all known results [1,10,11,14] as follows.…”

Section: Weighted Feedback Vertex Setmentioning

confidence: 99%

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Classifying Subset Feedback Vertex Set for $H$-Free Graphs

Paesani¹,

Paulusma²,

Rzążewski³

2022

Preprint

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In the Feedback Vertex Set problem, we aim to find a small set S of vertices in a graph intersecting every cycle. The Subset Feedback Vertex Set problem requires S to intersect only those cycles that include a vertex of some specified set T . We also consider the Weighted Subset Feedback Vertex Set problem, where each vertex u has weight w(u) > 0 and we ask that S has small weight. By combining known NP-hardness results with new polynomial-time results we prove full complexity dichotomies for Subset Feedback Vertex Set and Weighted Subset Feedback Vertex Set for H-free graphs, that is, graphs that do not contain a graph H as an induced subgraph.

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Section: Weighted Feedback Vertex Setmentioning

confidence: 99%

“…(the girth of a graph is the length of its shortest cycle). To give another relevant example, Feedback Vertex Set is also NP-complete for line graphs [10].…”

Section: Feedback Vertex Setmentioning

confidence: 99%

Classifying Subset Feedback Vertex Set for $H$-Free Graphs

Paesani¹,

Paulusma²,

Rzążewski³

2022

Preprint

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“…We first prove that Near-Bipartiteness is NP-complete for line graphs. It was already known that Feedback Vertex Set is NP-complete for line graphs of planar cubic bipartite graphs [37].…”

Section: Hardness When H Contains a Cycle Or Clawmentioning

confidence: 99%

“…It follows from Lemma 1 that if Feedback Vertex Set is NP-complete for some graph class G, then it is also NP-complete for G s . By starting from the fact that Feedback Vertex Set is NP-complete for line graphs of planar cubic bipartite graphs [37] and applying this observation a sufficient number of times, we find that for any constant g ≥ 3, Feedback Vertex Set is NP-complete for graphs of maximum degree at most 4 and girth at least g. Moreover, any non-independent feedback vertex set S of a subdivided copy G s of a graph G contains two adjacent vertices, one of which has degree 2 in G s . Hence, we can remove such a degree 2 vertex from S to obtain a smaller feedback vertex set of G s .…”

Section: Claim 2 the Graph G Has A Hamilton Path From V 1 To V 2 If Amentioning

confidence: 99%

Independent Feedback Vertex Set for $$P_5$$ P 5 -Free Graphs

et al. 2018

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The NP-complete problem Feedback Vertex Set is that of deciding whether or not it is possible, for a given integer k ≥ 0, to delete at most k vertices from a given graph so that what remains is a forest. The variant in which the deleted vertices must form an independent set is called Independent Feedback Vertex Set and is also NP-complete. In fact, even deciding if an independent feedback vertex set exists is NP-complete and this problem is closely related to the 3-Colouring problem, or equivalently, to the problem of deciding whether or not a graph has an independent odd cycle transversal, that is, an independent set of vertices whose deletion makes the graph bipartite. We initiate a systematic study of the complexity of Independent Feedback Vertex Set for H-free graphs. We prove that it is NP-complete if H contains a claw or cycle. Tamura, Ito and Zhou proved that it is polynomial-time solvable for P 4-free graphs. We show that it remains polynomial-time solvable for P 5-free graphs. We prove analogous results for the Independent Odd Cycle Transversal problem, which asks whether or not a graph has an independent odd cycle transversal of size at most k for a given integer k ≥ 0. Finally, in line with our underlying research aim, we compare the complexity of Independent Feedback Vertex Set for H-free graphs with the complexity of 3-Colouring, Independent Odd Cycle Transversal and other related problems. Keywords Independent feedback vertex set • Near-bipartiteness • Independent odd cycle transversal • 3-Colouring This paper received support from EPSRC (EP/K025090/1), London Mathematical Society (41536), the Leverhulme Trust (RPG-2016-258) and Fondation Sciences Mathématiques de Paris.

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“…Several other vertex partitioning problems on H-free graphs stay NP-complete as well if H has a cycle or claw. Examples of such problems include (Independent) Feedback Vertex Set [6,24,27], (Independent) Odd Cycle Transversal [6,10] and Even Cycle Transversal [25]. Hence, for all these problems, if H is a cycle or claw, then we need to add more structure to the class of input graphs in order to find tractable results for H-free graphs.…”

Section: Introductionmentioning

confidence: 99%

Partitioning H-Free Graphs of Bounded Diameter

Brause¹,

Golovach²,

Martin³

et al. 2021

Preprint

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A natural way of increasing our understanding of NP-complete graph problems is to restrict the input to a special graph class. Classes of H-free graphs, that is, graphs that do not contain some graph H as an induced subgraph, have proven to be an ideal testbed for such a complexity study. However, if the forbidden graph H contains a cycle or claw, then these problems often stay NP-complete. A recent complexity study (MFCS 2019) on the k-Colouring problem shows that we may still obtain tractable results if we also bound the diameter of the H-free input graph. We continue this line of research by initiating a complexity study on the impact of bounding the diameter for a variety of classical vertex partitioning problems restricted to H-free graphs. We prove that bounding the diameter does not help for Independent Set, but leads to new tractable cases for problems closely related to 3-Colouring. That is, we show that Near-Bipartiteness, Independent Feedback Vertex Set, Independent Odd Cycle Transversal, Acyclic 3-Colouring and Star 3-Colouring are all polynomial-time solvable for chair-free graphs of bounded diameter. To obtain these results we exploit a new structural property of 3-colourable chair-free graphs.

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On line graphs of subcubic triangle-free graphs

Cited by 21 publications

References 34 publications

Classifying Subset Feedback Vertex Set for $H$-Free Graphs

Classifying Subset Feedback Vertex Set for $H$-Free Graphs

Independent Feedback Vertex Set for $$P_5$$ P 5 -Free Graphs

Partitioning H-Free Graphs of Bounded Diameter

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