On the Complexity of Claw-Free Vertex Splitting

TL;DR

This paper investigates the computational complexity of transforming graphs into claw-free graphs through vertex splitting, proving NP-completeness in general but polynomial-time solvability for graphs with maximum degree 4, and extending results to $K_{1,c}$-free cases.

Contribution

It establishes the NP-completeness of Claw-Free Exclusive Vertex Splitting and provides a polynomial-time algorithm for graphs with maximum degree 4, resolving an open problem.

Findings

NP-completeness of the problem in general

Polynomial-time algorithm for degree-4 graphs

Extension to $K_{1,c}$-free vertex splitting

Abstract

Vertex splitting consists of taking a vertex $v$ in a graph and replacing it with two non-adjacent vertices whose combined neighborhoods is the neighborhood of $v$. The split is said to be exclusive when these neighborhoods are disjoint. In the Claw-Free (Exclusive) Vertex Splitting problem, we are given a graph $G$ and an integer $k$, and we are asked if we can perform at most $k$ (exclusive) vertex splits to obtain a claw-free graph. We consider the complexity of Claw-Free Exclusive Vertex Splitting and prove it to be NP-complete in general, while admitting a polynomial-time algorithm when the input graph has maximum degree 4. This result settles an open problem posed in [Firbas \& Sorge, ISAAC 2024]. We also show that our results can be generalized to $K_{1,c}$-Free Vertex Splitting for all $c \geq 3$.