Computer assisted discovery: Zero forcing vs vertex cover

TL;DR

This paper demonstrates how an automated conjecturing tool, TxGraffiti, was used to prove a relation between zero forcing and vertex cover numbers in claw-free graphs, providing polynomial algorithms and infinite graph families.

Contribution

It proves a conjecture relating zero forcing and vertex cover in claw-free graphs and extends the relation to broader classes of graphs with maximum degree at least 3.

Findings

Proved that for claw-free graphs, Z(G) ≥ β(G).

Developed a polynomial-time algorithm to find zero forcing sets of size β(G).

Established a bound Z(G) ≤ (Δ−2)β(G)+1 for connected graphs with Δ ≥ 3.

Abstract

In this paper, we showcase the process of using an automated conjecturing program called \emph{TxGraffiti} written and maintained by the second author. We begin by proving a conjecture formulated by \emph{TxGraffiti} that for a claw-free graph $G$, the vertex cover number $\beta(G)$ is greater than or equal to the zero forcing number $Z(G)$. Our proof of this result is constructive, and yields a polynomial time algorithm to find a zero forcing set with cardinality $\beta(G)$. We also use the output of \emph{TxGraffiti} to construct several infinite families of claw-free graphs for which $Z(G)=\beta(G)$. Additionally, inspired by the aforementioned conjecture of \emph{TxGraffiti}, we also prove a more general relation between the zero forcing number and the vertex cover number for any connected graph with maximum degree $\Delta \ge 3$, namely that $Z(G)\leq (\Delta-2)\beta(G)$+1.