Graphs with Extremal Connected Forcing Numbers

TL;DR

This paper characterizes graphs with specific connected forcing numbers, extending previous zero forcing results using combinatorial methods and providing new structural insights into connected forcing sets.

Contribution

It extends the characterization of graphs with connected forcing numbers 2 and n-2, using combinatorial techniques instead of linear algebra.

Findings

Characterization of graphs with connected forcing number 2

Characterization of graphs with connected forcing number n-2

Structural results on connected forcing sets

Abstract

Zero forcing is an iterative graph coloring process where at each discrete time step, a colored vertex with a single uncolored neighbor forces that neighbor to become colored. The zero forcing number of a graph is the cardinality of the smallest set of initially colored vertices which forces the entire graph to eventually become colored. Connected forcing is a variant of zero forcing in which the initially colored set of vertices induces a connected subgraph; the analogous parameter of interest is the connected forcing number. In this paper, we characterize the graphs with connected forcing numbers 2 and $n-2$. Our results extend existing characterizations of graphs with zero forcing numbers 2 and $n-2$; we use combinatorial and graph theoretic techniques, in contrast to the linear algebraic approach used to obtain the latter. We also present several other structural results about the connected forcing sets of a graph.