On the zero forcing number of corona and lexicographic product of graphs

TL;DR

This paper investigates the zero forcing number for corona and lexicographic graph products, deriving formulas and bounds that relate the zero forcing number of these products to their component graphs.

Contribution

It provides new explicit formulas and bounds for the zero forcing number of corona and lexicographic graph products, extending understanding of these graph operations.

Findings

Zero forcing number of iterated corona products is expressed recursively.

Bounds for the zero forcing number of the lexicographic product are established.

Formulas relate the zero forcing number to component graph properties.

Abstract

The zero forcing number of a graph $G$, denoted by $Z(G)$, is the minimum cardinality of a set $S$ of black vertices (where vertices in $V(G)\setminus S$ are colored white) such that $V(G)$ is turned black after finitely many applications of $"$the color change rule$"$: a white vertex is turned black if it is the only white neighbor of a black vertex. In this paper, we study the zero forcing number of corona product, $G\odot H$ and lexicographic product, $G\circ H$ of two graphs $G$ and $H$. It is shown that if $G$ and $H$ are connected graphs of order $n_{1}\geq2$ and $n_{2}\geq2$ respectively, then $Z(G\odot ^{k}H)=Z(G\odot ^{k-1}H)+n_{1}(n_{2}+1)^{k-1}Z(H)$, where $G\odot^{k}H=(G\odot^{k-1}H)\odot H$. Also, it is shown that for a connected graph $G$ of order $n\geq 2$ and an arbitrary graph $H$ containing $l\geq 1$ components $H_{1},H_{2}, \cdots,H_{l}$ with $|V(H_{i})|=m_{i}\geq 2$, $1\leq i\leq l$, $(n-1)l+\sum\limits_{i=1}^l m_{i}\leq Z(G\circ H)\leq n(\sum\limits_{i=1}^{l}m_{i})-l$.