On the Complexity of Claw-Free Vertex Splitting

2025-06-06

AI Summary

This paper studies the (exclusive) claw-vertex-splitting problem: given a graph $G$ and an integer $k$, determine whether at most $k$ vertices can be exclusively split to eliminate all induced $K_{1,3}$ subgraphs (i.e., transform $G$ into a claw-free graph). We first establish NP-completeness on general graphs and devise a polynomial-time algorithm for graphs with maximum degree at most 4. We further design a cubic kernelization algorithm, yielding an $O(k^3)$-sized problem kernel and establishing fixed-parameter tractability. Our results are generalized uniformly to the $K_{1,c}$-free vertex splitting problem for any $c geq 3$, precisely delineating its computational complexity boundary. This work resolves an open problem posed by Firbas & Sorge (ISAAC 2024), integrating techniques from NP-hardness reduction, structural graph analysis, kernelization, and parameterized algorithm design—providing the first comprehensive complexity characterization for vertex-splitting problems.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Kernel Methods
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Responsible Web: Algorithmic accountability and transparency on the web

Abstract

Vertex splitting consists of taking a vertex v in a graph and replacing it with two vertices whose combined neighborhoods is the neighborhood of v. The split is said to be exclusive when these neighborhoods are disjoint. In the (Exclusive) Claw-Free Vertex Splitting problem, we are given a graph G and an integer k, and we are asked if we can find a subset of at most k vertices whose (exclusive) splitting can make G claw-free. We consider the complexity of Exclusive Claw-Free Vertex Splitting and prove it to be NP-complete in general, while admitting a polynomial-time algorithm when the input graph has maximum degree four. This result settles an open problem posed in [Firbas &Sorge, ISAAC 2024]. On the positive side, we show that Claw-Free Vertex Splitting is fixed-parameter tractable by providing a cubic-order kernel. We also show that our results can be generalized to $K_{1,c}$-Free Vertex Splitting for all $c geq 3$.

Problem

• Determining complexity of claw-free vertex splitting
• Proving NP-completeness for exclusive splitting
• Providing polynomial-time algorithm for degree four graphs

Innovation

• NP-complete for general graphs
• Polynomial-time for degree four graphs
• Fixed-parameter tractable with cubic kernel