Fault Tolerant Zero Forcing
Asher Brown, Mark Hunnell, Za'Kiyah Toomer-Sanders, Sarah Weber
TL;DR
This work introduces the $k$-fault tolerant zero forcing number, $Z_t^k(G)$, a robustness variant of the standard zero forcing parameter $Z(G)$, defined as the minimum $|B|$ such that every subset of $B$ of size $|B|-k$ is a zero forcing set. It establishes a general lower bound $Z_t^k(G)\ge Z(G)+k$, explores existence (noting that $Z_t^k(G)$ may fail to exist for $k\ge 2$ in some graphs) and computes exact values for multiple graph families, including $P_n$, $K_n$, $K_{1,n}$, $K_m,n$, $C_n$, and $W_n$. A central contribution is the development of compatible collections of path covers and the irredundance concept $I(G)$, yielding $I(G)\le Z_t(G)$, with a complete characterization for trees: $Z_t(T)=I(T)$. The paper also studies the behavior of $Z_t$ under common graph operations and discusses the role of twins, providing structural insights into robustness of the zero forcing process and highlighting when $Z_t^k(G)$ exists or fails to.
Abstract
Zero forcing is an iterative graph coloring process studied for its wide array of applications. In this process, the vertices of the graph are initially designated as blue or white, and a zero forcing set is a set of initially blue vertices that results in all vertices becoming blue after repeated application of a color change rule. The zero forcing number of a graph is the minimum cardinality of a zero forcing set. The zero forcing number has motivated the introduction of a host of variants motivated by linear-algebraic or graph-theoretic contexts. We define a variant we term the $k$-fault tolerant zero forcing number, which is the minimum cardinality of a set $B$ such that every subset of $B$ of cardinality $|B|-k$ is a zero forcing set. We study the values of this parameter on various graph families, the behavior under several graph operations, and characterize the 1-fault tolerant zero forcing number of trees.