Throttling for standard zero forcing on directed graphs

TL;DR

This paper explores the concept of throttling in zero forcing on directed graphs, characterizing graphs with low throttling numbers, analyzing effects of arc flips and vertex deletions, and introducing the orientation throttling interval for undirected graphs.

Contribution

It introduces the orientation throttling interval (OTI) for undirected graphs and characterizes digraphs with low throttling numbers, advancing understanding of zero forcing dynamics.

Findings

Characterized all simple digraphs with throttling number ≤ t.

Analyzed how arc flips and vertex deletions affect throttling.

Proposed the orientation throttling interval (OTI) and provided bounds.

Abstract

Zero forcing is a process on graphs in which a color change rule is used to force vertices to become blue. The amount of time taken for all vertices in the graph to become blue is the propagation time. Throttling minimizes the sum of the number of initial blue vertices and the propagation time. In this paper, we study throttling in the context of directed graphs (digraphs). We characterize all simple digraphs with throttling number at most $t$ and examine the change in the throttling number after flipping arcs and deleting vertices. We also introduce the orientation throttling interval (OTI) of an undirected graph, which is the range of throttling numbers achieved by the orientations of the graph. While the OTI is shown to vary among different graph families, some general bounds are obtained. Additionally, the maximum value of the OTI of a path is conjectured to be achieved by the orientation of a path whose arcs alternate in direction. The throttling number of this orientation is exactly determined in terms of the number of vertices.