Fractional Zero Forcing via Three-color Forcing Games

TL;DR

This paper introduces a fractional positive semidefinite zero forcing process using three-color forcing games, providing new methods to compute and analyze the fractional positive semidefinite forcing number and its relation to skew zero forcing.

Contribution

It develops a novel three-color forcing game to directly compute the fractional positive semidefinite forcing number and characterizes graphs with zero skew zero forcing number.

Findings

Introduces a fractional zero forcing parameter based on three-color forcing.

Shows the fractional parameter equals the skew zero forcing number.

Provides an algorithm to identify graphs with zero skew zero forcing number.

Abstract

An $r$-fold analogue of the positive semidefinite zero forcing process that is carried out on the $r$-blowup of a graph is introduced and used to define the fractional positive semidefinite forcing number. Properties of the graph blowup when colored with a fractional positive semidefinite forcing set are examined and used to define a three-color forcing game that directly computes the fractional positive semidefinite forcing number of a graph. We develop a fractional parameter based on the standard zero forcing process and it is shown that this parameter is exactly the skew zero forcing number with a three-color approach. This approach and an algorithm are used to characterize graphs whose skew zero forcing number equals zero.