This paper begins the study of reconfiguration zero forcing sets, and more specifically, graph. Given a base graph G, its graph, Z(G), is whose vertices are minimum sets G with an edge between B B′ Z(G) if only can be obtained from by changing single vertex G. It shown that forest connected, but many graphs disconnected. We characterize complete graphs, show star cannot computing takes 2Θ(n) op...

We prove that a number of axioms, each a consequence of PFA (the Proper Forcing Axiom) are equivalent. In particular we show that TOP (the Thinning-out Principle as introduced by Baumgartner in the Handbook of set-theoretic topology), is equivalent to the following statement: If I is an ideal on co, with co, generators, then there exists an uncountable X C co,, such that either [X]w n I = 0 or ...

The zero forcing number Z(G) is used to study the minimum rank/maximum nullity of the family of symmetric matrices described by a simple, undirected graph G. The positive semidefinite zero forcing number is a variant of the (standard) zero forcing number, which uses the same definition except with a different color-change rule. The positive semidefinite maximum nullity and zero forcing number f...

Given a simple undirected graph G and a positive integer k, the k-forcing number of G, denoted Fk(G), is the minimum number of vertices that need to be initially colored so that all vertices eventually become colored during the discrete dynamical process described by the following rule. Starting from an initial set of colored vertices and stopping when all vertices are colored: if a colored ver...

Let $G=(V,E)$ be a simple connected graph. A perfect matching (or Kekul'e structure in chemical literature) of $G$ is a set of disjoint edges which covers all vertices of $G$. The anti-forcing number of $G$ is the smallest number of edges such that the remaining graph obtained by deleting these edges has a unique perfect matching and is denoted by $af(G)$. In this paper we consider some specifi...

The zero forcing number of a graph G is the cardinality of the smallest subset of the vertices of G that forces the entire graph using a color change rule. This paper presents some basic properties of circulant graphs and later investigates zero forcing numbers of circulant graphs of the form C[n, {s, t}], while also giving attention to propagation time for specific zero forcing sets.

The minimum rank of a simple graph G is defined to be the smallest possible rank over all symmetric real matrices whose ijth entry (for i = j) is nonzero whenever {i, j} is an edge in G and is zero otherwise. This paper introduces a new graph parameter, Z(G), that is the minimum size of a zero forcing set of vertices and uses it to bound the minimum rank for numerous families of graphs, often e...