The metric dimension dim(G) of a graph G is the minimum number of vertices such that every vertex of G is uniquely determined by its vector of distances to the chosen vertices. The 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)\S are colored white) such that V (G) is converted entirely to black after finitely many applica...

We show that certain types of zero-forcing sets for a graph give rise to chordal supergraphs and hence to proper colorings. Zero-forcing was originally defined to provide a bound for matrix minimum rank problems [1], but is interesting as a graph-theoretic notion in its own right [5], and has applications to mathematical physics, such as quantum systems [3]. There are different flavors of zero-...

Zero forcing is a propagation process on a graph, or digraph, defined in linear algebra to provide a bound for the minimum rank problem. Independently, zero forcing was introduced in physics, computer science and network science, areas where line digraphs are frequently used as models. Zero forcing is also related to power domination, a propagation process that models the monitoring of electric...

Abstract. Let G be a simple, undirected graph. Positive semidefinite (PSD) zero forcing on G is based on the following 1 color-change rule: Let W1,W2, . . . ,Wk be the sets of vertices of the k connected components in G − B (where B is a set of blue 2 vertices). If w ∈Wi is the only white neighbor of some b ∈ B in the graph G[B∪Wi], then we change w to blue. A minimum positive 3 semidefinite ze...

A subset S of initially infected vertices of a graph G is called forcing if we can infect the entire graph by iteratively applying the following process. At each step, any infected vertex which has a unique uninfected neighbour, infects this neighbour. The forcing number of G is the minimum cardinality of a forcing set in G. In the present paper, we study the forcing number of various classes o...

Using the theory of rudimentary recursion and provident sets developed in a previous paper, we give a treatment of set forcing appropriate for working over models of a theory PROVI which may plausibly claim to be the weakest set theory supporting a smooth theory of set forcing, and of which the minimal model is Jensen’s Jω. Much of the development is rudimentary or at worst given by rudimentary...