Planarizing Graphs and Their Drawings by Vertex Splitting

Nöllenburg, Martin; Sorge, Manuel; Terziadis, Soeren; Villedieu, Anaïs; Wu, Hsiang‐Yun; Wulms, Jules

doi:10.1007/978-3-031-22203-0_17

Cited by 6 publications

(3 citation statements)

References 37 publications

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“…Nöllenburg et al [17] showed that, for any minor-closed graph class Π, the graph class Π k containing all graphs that can be modified to a graph in Π using at most k vertex splits is also minor-closed. Robertson and Seymour [21] showed that every minor-closed graph class has a constant-size set of forbidden minors and that it can be tested in cubic time whether a graph contains a given fixed graph as a minor.…”

Section: Vertex Splittingmentioning

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: Vertex Splittingmentioning

confidence: 99%

“…Nöllenburg et al [17] have recently studied the vertex splitting problem, which is known to be NP-complete [11]. In particular, they gave a non-uniform FPT-algorithm for deciding whether a given graph can be planarized with at most k splits.…”

Section: Introductionmentioning

confidence: 99%

Parameterized Complexity of Vertex Splitting to Pathwidth at most 1

Baumann¹,

Pfretzschner²,

Rutter³

2023

Preprint

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show abstract

“…In bipartite graphs arising in domain applications, such as visualizing relationships between anatomical structures and cell types in the human body [7], vertex splitting makes sense only on one side of the layout. Several variants of optimizing such layouts have been recently studied [2,20,3].…”

Section: Introductionmentioning

confidence: 99%

Visualization of Bipartite Graphs in Limited Window Size

Efrat,

Evans,

Köck

et al. 2024

Preprint

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Bipartite graphs are commonly used to visualize objects and their features. An object may possess several features and several objects may share a common feature.The standard visualization of bipartite graphs, with objects and features on two (say horizontal) parallel lines at integer coordinates and edges drawn as line segments, can often be difficult to work with. A common task in visualization of such graphs is to consider one object and all its features. This naturally defines a drawing window, defined as the smallest interval that contains the x-coordinates of the object and all its features. We show that if both objects and features can be reordered, minimizing the average window size is NP-hard. However, if the features are fixed, then we provide an efficient polynomial time algorithm for arranging the objects, so as to minimize the average window size. Finally, we introduce a different way of visualizing the bipartite graph, by placing the nodes of the two parts on two concentric circles. For this setting we also show NP-hardness for the general case and a polynomial time algorithm when the features are fixed.

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