Probabilistic Zero Forcing in Graphs

Abstract

The \emph{zero forcing number} $Z(G)$ of a graph $G$ is the minimum cardinality of a set $S$ of black vertices (whereas vertices in $V(G)\setminusS$ are colored white) such that $V(G)$ is turned black after finitely many applications of "the (classical) color change rule": a white vertex is converted to a black vertex if it is the only white neighbor of a black vertex. Zero forcing number was introduced and used to bound the minimum rank of graphs by the "AIM Minimum Rank - Special Graphs Work Group". We introduce here a probabilistic color change rule (pccr) which is a natural generalization of the classical color change rule. We introduce a theory of probabilistic zero forcing arising out of the pccr; the theory yields a quantity $P_A(G)$, which can be viewed as the probability that a graph $G$ with an initial black set $A$ will be converted entirely to the color black. We also interpret the evolution of the sample spaces of this theory as a Markov process. We end with a few basic examples illustrating this theory.