An FPT Algorithm for Bipartite Vertex Splitting

Ahmed, Reyan; Kobourov, Stephen G.; Kryven, Myroslav

doi:10.1007/978-3-031-22203-0_19

Cited by 6 publications

(5 citation statements)

References 17 publications

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“…For POVE, we gave an O(4 k • m)-time branching algorithm and showed that POVE admits a quadratic kernel that can be computed in linear time. This improves on a recent result by Ahmed et al [2], who developed a kernel of size O(k 6 ) for a more restricted version of the problem.…”

Section: Discussionsupporting

confidence: 78%

“…Second, in Section 4, we show that POVS has a kernel of size 16k and it admits an algorithm with running time O((6k + 12) k • m). This answers the open question of Ahmed et al [2].…”

Section: Pathwidth-one Vertex Explosion (Pove) Inputsupporting

confidence: 67%

“…First, in Section 3, we show that POVE admits a kernel of size O(k 2 ) and an algorithm with running time O(4 k m), thereby improving over the results of Ahmed et al [2] in a more general setting.…”

Section: Pathwidth-one Vertex Explosion (Pove) Inputmentioning

confidence: 61%

“…Ahmed et al [2] investigated the problem of splitting the vertices of a bipartite graph so that it admits a 2-layered drawing without crossings. They assume that the input graph is bipartite and only the vertices of one of the two sets in the bipartition may be split.…”

Section: Introductionmentioning

confidence: 99%

“…Motivated by this, we more generally consider the problem of turning a graph G = (V, E) into a graph of pathwidth at most 1 by the above operations. In order to model the restriction of Ahmed et al [2] that only one side of their bipartite input graph may be split, we further assume that we are given a subset S ⊆ V , to which we may apply modification operations as part of the input.…”

Section: Introductionmentioning

confidence: 99%

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Parameterized Complexity of Vertex Splitting to Pathwidth at most 1

Baumann¹,

Pfretzschner²,

Rutter³

2023

Preprint

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Motivated by the planarization of 2-layered straight-line drawings, we consider the problem of modifying a graph such that the resulting graph has pathwidth at most 1. The problem Pathwidth-One Vertex Explosion (POVE) asks whether such a graph can be obtained using at most k vertex explosions, where a vertex explosion replaces a vertex v by deg(v) degree-1 vertices, each incident to exactly one edge that was originally incident to v. For POVE, we give an FPT algorithm with running time O( 4k • m) and a quadratic kernel, thereby improving over the O(k 6 )-kernel by Ahmed et al. [2] in a more general setting. Similarly, a vertex split replaces a vertex v by two distinct vertices v1 and v2 and distributes the edges originally incident to v arbitrarily to v1 and v2. Analogously to POVE, we define the problem variant Pathwidth-One Vertex Splitting (POVS) that uses the split operation instead of vertex explosions. Here we obtain a linear kernel and an algorithm with running time O((6k + 12) k • m). This answers an open question by Ahmed et al. [2].Finally, we consider the problem Π Vertex Splitting (Π-VS), which generalizes the problem POVS and asks whether a given graph can be turned into a graph of a specific graph class Π using at most k vertex splits. For graph classes Π that can be tested in monadic second-order graph logic (MSO2), we show that the problem Π-VS can be expressed as an MSO2 formula, resulting in an FPT algorithm for Π-VS parameterized by k if Π additionally has bounded treewidth. We obtain the same result for the problem variant using vertex explosions.

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Section: Discussionsupporting

confidence: 78%

“…Second, in Section 4, we show that POVS has a kernel of size 16k and it admits an algorithm with running time O((6k + 12) k • m). This answers the open question of Ahmed et al [2].…”

Section: Pathwidth-one Vertex Explosion (Pove) Inputsupporting

confidence: 67%

Section: Pathwidth-one Vertex Explosion (Pove) Inputmentioning

confidence: 61%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations