Properties of a $q$-analogue of zero forcing

TL;DR

This paper explores a $q$-analogue of zero forcing on graphs, introducing an oracle-based option, and characterizes minimal costs and zero forcing numbers for specific graph classes and parameters.

Contribution

It introduces a new $q$-analogue of zero forcing, analyzes its properties, and determines minimal costs and zero forcing numbers for certain graphs and parameters.

Findings

Identifies graphs with minimal cost 1 or 2 for all $q$

Determines zero forcing number for all trees when $q=1$

Establishes basic properties of the $q$-analogue game

Abstract

Zero forcing is a combinatorial game played on a graph where the goal is to start with all vertices unfilled and to change them to filled at minimal cost. In the original variation of the game there were two options. Namely, to fill any one single vertex at the cost of a single token; or if any currently filled vertex has a unique non-filled neighbor, then the neighbor is filled for free. This paper investigates a $q$-analogue of zero forcing which introduces a third option involving an oracle. Basic properties of this game are established including determining all graphs which have minimal cost $1$ or $2$ for all possible $q$, and finding the zero forcing number for all trees when $q=1$.