A Characterization of Claw-Free Graphs using Zero Forcing Invariants

TL;DR

This paper proves that for claw-free graphs, the standard and positive semidefinite zero forcing numbers are equal, confirming a conjecture and establishing a new characterization of claw-free graphs based on these invariants.

Contribution

It establishes that Z(G) equals Z_+(G) for all claw-free graphs, resolving a conjecture and providing a new graph characterization.

Findings

Z(G) = Z_+(G) for all claw-free graphs

Claw-free graphs characterized by equal zero forcing numbers in all induced subgraphs

Confirmed conjecture from TxGraffiti program

Abstract

We prove that the \emph{standard zero forcing number} $Z(G)$ and the \emph{positive semidefinite zero forcing number} $Z_+(G)$ are equal for all claw-free graphs $G$. This result resolves a conjecture proposed by the computer program \emph{TxGraffiti} and highlights a connection between these graph invariants in claw-free structures. As a corollary, we show that a graph $G$ is claw-free if and only if every induced subgraph $H \subseteq G$ satisfies $Z(H) = Z_+(H)$.