TY - GEN

T1 - Planarizing Graphs and Their Drawings by Vertex Splitting

AU - Nöllenburg, Martin

AU - Sorge, Manuel

AU - Terziadis, Soeren

AU - Villedieu, Anaïs

AU - Wu, Hsiang Yun

AU - Wulms, Jules

N1 - Publisher Copyright:
© 2023, The Author(s), under exclusive license to Springer Nature Switzerland AG.

PY - 2023/1/1

Y1 - 2023/1/1

N2 - The splitting number of a graph G= (V, E) is the minimum number of vertex splits required to turn G into a planar graph, where a vertex split removes a vertex v∈ V, introduces two new vertices v1, v2, and distributes the edges formerly incident to v among v1, v2. The splitting number problem is known to be NP-complete for abstract graphs and we provide a non-uniform fixed-parameter tractable (FPT) algorithm for this problem. We then shift focus to the splitting number of a given topological graph drawing in R2, where the new vertices resulting from vertex splits must be re-embedded into the existing drawing of the remaining graph. We show NP-completeness of this embedded splitting number problem, even for its two subproblems of (1) selecting a minimum subset of vertices to split and (2) for re-embedding a minimum number of copies of a given set of vertices. For the latter problem we present an FPT algorithm parameterized by the number of vertex splits. This algorithm reduces to a bounded outerplanarity case and uses an intricate dynamic program on a sphere-cut decomposition.

AB - The splitting number of a graph G= (V, E) is the minimum number of vertex splits required to turn G into a planar graph, where a vertex split removes a vertex v∈ V, introduces two new vertices v1, v2, and distributes the edges formerly incident to v among v1, v2. The splitting number problem is known to be NP-complete for abstract graphs and we provide a non-uniform fixed-parameter tractable (FPT) algorithm for this problem. We then shift focus to the splitting number of a given topological graph drawing in R2, where the new vertices resulting from vertex splits must be re-embedded into the existing drawing of the remaining graph. We show NP-completeness of this embedded splitting number problem, even for its two subproblems of (1) selecting a minimum subset of vertices to split and (2) for re-embedding a minimum number of copies of a given set of vertices. For the latter problem we present an FPT algorithm parameterized by the number of vertex splits. This algorithm reduces to a bounded outerplanarity case and uses an intricate dynamic program on a sphere-cut decomposition.

KW - Parameterized complexity

KW - Planarization

KW - Vertex splitting

UR - http://www.scopus.com/inward/record.url?scp=85148693423&partnerID=8YFLogxK

U2 - 10.1007/978-3-031-22203-0_17

DO - 10.1007/978-3-031-22203-0_17

M3 - Conference contribution

AN - SCOPUS:85148693423

SN - 9783031222023

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 232

EP - 246

BT - Graph Drawing and Network Visualization - 30th International Symposium, GD 2022, Revised Selected Papers

A2 - Angelini, Patrizio

A2 - von Hanxleden, Reinhard

PB - Springer Science and Business Media Deutschland GmbH

T2 - 30th International Symposium on Graph Drawing and Network Visualization, GD 2022

Y2 - 13 September 2022 through 16 September 2022

ER -