Effects of vertex degrees on the zero-forcing number and propagation time of a graph

Abstract

In this paper, we present new bounds on the zero-forcing propagation time and zero-forcing number. Zero-forcing is based on iterating the following color change rule. If a vertex is colored and has only one neighbor that is not colored, then that uncolored neighbor becomes colored in the next iteration. The zero-forcing number is the size of a minimum set of vertices such that repeated iterations of the color change rule will color the entire graph. The zero-forcing propagation time is the minimum number of iterations required to color the entire graph from a minimum zero-forcing set. We present new bounds on the zero-forcing propagation time for cubic graphs. These bounds reduce the known bounds that are based on the minimum degree of the graph. We introduce zero-forcing forts, a special type of subgraph that must contain a vertex in any minimum zero-forcing set, and use these forts to bound the zero-forcing number of a graph. We also show that the zero-forcing number of a graph is bounded from below by its branchwidth.