On Zero Forcing Number of Functigraphs

TL;DR

This paper investigates the zero forcing number of functigraphs, establishing bounds based on the minimum degree of the original graph and exploring specific cases like complete graphs, cycles, and paths.

Contribution

It introduces bounds for the zero forcing number of functigraphs for any function and analyzes specific graph classes, expanding understanding of zero forcing in complex graph constructions.

Findings

Bounds established: 1+δ(G) ≤ Z(C(G,f)) ≤ 2n-2 for any function f

No universal function g relates Z(G) and Z(C(G,f)) in a consistent way

Zero forcing numbers for functigraphs on complete graphs, cycles, and paths analyzed

Abstract

\emph{Zero forcing number}, $Z(G)$, of a graph $G$ is the minimum cardinality of a set $S$ of black vertices (whereas vertices in $V(G) \setminus S$ are colored white) such that $V(G)$ is turned black after finitely many applications of "the color-change rule": a white vertex is converted black 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". Let $G_1$ and $G_2$ be disjoint copies of a graph $G$ and let $f: V(G_1) \rightarrow V(G_2)$ be a function. Then a \emph{functigraph} $C(G, f)=(V, E)$ has the vertex set $V=V(G_1) \cup V(G_2)$ and the edge set $E=E(G_1) \cup E(G_2) \cup \{uv \mid v=f(u)\}$. For a connected graph $G$ of order $n \ge 3$, it is readily seen that $1+\delta(G) \le Z(C(G, \sigma)) \le n$ for any permutation $\sigma$; we show that $1+ \delta(G) \le Z(C(G, f)) \le 2n-2$ for any function $f$, where $\delta(G)$ is the minimum degree of $G$. We give examples showing that there does not exist a function $g$ such that, for every pair $(G,f)$, $Z(G)<g(Z(C(G,f)))$ or $g(Z(G))>Z(C(G,f))$. We further investigate the zero forcing number of functigraphs on complete graphs, on cycles, and on paths.