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On the Complexity of Establishing Hereditary Graph Properties via Vertex Splitting

Authors Alexander Firbas , Manuel Sorge

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  • Theory of computation → Problems, reductions and completeness
Keywords
  • NP-completeness
  • polynomial-time solvability
  • graph theory
  • graph transformation
  • graph modification

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Abstract

Vertex splitting is a graph operation that replaces a vertex v with two nonadjacent new vertices u, w and makes each neighbor of v adjacent with one or both of u or w. Vertex splitting has been used in contexts from circuit design to statistical analysis. In this work, we generalize from specific vertex-splitting problems and systematically explore the computational complexity of achieving a given graph property Π by a limited number of vertex splits, formalized as the problem Π Vertex Splitting (Π-VS). We focus on hereditary graph properties and contribute four groups of results: First, we classify the classical complexity of Π-VS for graph properties characterized by forbidden subgraphs of order at most 3. Second, we provide a framework that allows one to show NP-completeness whenever one can construct a combination of a forbidden subgraph and prescribed vertex splits that satisfy certain conditions. Using this framework we show NP-completeness when Π is characterized by sufficiently well-connected forbidden subgraphs. In particular, we show that F-Free-VS is NP-complete for each biconnected graph F. Third, we study infinite families of forbidden subgraphs, obtaining NP-completeness for Bipartite-VS and Perfect-VS, contrasting the known result that Π-VS is in P if Π is the set of all cycles. Finally, we contribute to the study of the parameterized complexity of Π-VS with respect to the number of allowed splits. We show para-NP-hardness for K₃-Free-VS and derive an XP-algorithm when each vertex is only allowed to be split at most once, showing that the ability to split a vertex more than once is a key driver of the problems' complexity.

Cite As Get BibTex

Alexander Firbas and Manuel Sorge. On the Complexity of Establishing Hereditary Graph Properties via Vertex Splitting. In 35th International Symposium on Algorithms and Computation (ISAAC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 322, pp. 30:1-30:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ISAAC.2024.30

Author Details

Alexander Firbas
  • TU Wien, Austria
Manuel Sorge
  • TU Wien, Austria

Funding

Alexander Firbas acknowledges support from the Vienna Science and Technology Fund (WWTF) [10.47379/ICT22029]. Manuel Sorge acknowledges partial support from the Alexander von Humboldt Foundation.

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