Hardness of Vertex Splitting: Cographs, Chordal Graphs, and Beyond

2026-07-15

AI Summary

This study investigates the computational complexity of transforming a graph into specific graph classes—such as cographs, chordal graphs, and unit interval graphs—via vertex splitting operations. It establishes, for the first time, that the vertex splitting problem is NP-complete for each of these classes under a bound on the number of allowed splits, thereby resolving several open questions. Under the Exponential Time Hypothesis, the work further derives fine-grained lower bounds, showing that no algorithms running in time $2^{o(k)}n^{O(1)}$ or $2^{o(n)}$ exist for cographs and chordal graphs (though not for unit interval graphs). Combining complexity theory, reduction techniques, and graph-theoretic analysis, the paper systematically delineates the intractability landscape of vertex splitting across multiple graph classes, including variants such as exclusive and shallow splitting.

Abstract

Vertex splitting replaces a vertex (v) by two nonadjacent vertices whose neighborhoods together equal (N(v)). A split is \emph{exclusive} if these neighborhoods are disjoint and \emph{shallow} if no newly created vertex is split again. For a graph property (Π), \textsc{(Π)-Vertex Splitting} asks whether at most (k) splits can transform a graph (G) into one satisfying (Π). We continue the systematic study of this operation and settle several open problems. First, we prove that \textsc{Cograph Vertex Splitting} is \textsf{NP}-complete, even on graphs of girth at least 5, resolving a question of Firbas and Sorge (ISAAC 2024). More generally, \textsc{(P_t)-free Vertex Splitting} is \textsf{NP}-complete for every fixed (t\geq 4). We also prove that \textsc{Chordal Vertex Splitting} and \textsc{Unit-Interval Vertex Splitting} are \textsf{NP}-complete, resolving two questions of Abu-Khzam, Chakraborty, Isenmann, and Oijid (IWOCA 2026). Our hardness results extend to the exclusive and shallow variants. Assuming the Exponential Time Hypothesis, none of these problems admits an algorithm running in (2^{o(k)}n^{O(1)}) time; moreover, except for the unit-interval cases, none admits an algorithm running in (2^{o(n)}) time.

Problem

• Vertex Splitting
• Cograph
• Chordal Graph
• NP-completeness
• Graph Modification

Innovation

• Vertex Splitting
• NP-completeness
• Cographs
• Chordal Graphs
• Exponential Time Hypothesis