On the Complexity of Claw-Free Vertex Splitting

TL;DR

The complexity of Claw-Free Exclusive Vertex Splitting is considered and it is proved to be NP-complete in general, while admitting a polynomial-time algorithm when the input graph has maximum degree 4.

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 $c \geq 3$.

Figures (5)

• Figure 1: Case 1.
• Figure 2: Case 2.
• Figure 3: Case 3.
• Figure 4: Case 4.
• Figure 5: Construction for the instance where the given graph is a $K_4$.

Theorems & Definitions (6)

• theorem 3
• proof
• theorem 4
• proof
• corollary 1
• corollary 2