Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Throttling for standard zero forcing on directed graphs

Emily Cairncross, Joshua Carlson, Peter Hollander, Benjamin Kitchen, Emily Lopez, Ashley Zhuang

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.

Throttling for standard zero forcing on directed graphs

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 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.

Paper Structure

This paper contains 8 sections, 26 theorems, 35 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Throttling for simple digraphs
  3. Monotonicity and characterizations
  4. Flipping arcs and other digraph operations
  5. Throttling for orientations of simple graphs
  6. Throttling on alternating paths
  7. Concluding Remarks
  8. Acknowledgements

Key Result

Theorem 2.6

Given a simple digraph $\Gamma$ and a positive integer $t$, $\operatorname{th} (\Gamma) \leq t$ if and only if there exist integers $a\geq 1$ and $b\geq 0$ such that $a+b = t$ and $\Gamma$ can be obtained from $H_{a,b+1}$ by contracting path arcs and deleting non-path arcs.

Figures (10)

  • Figure 1: An oriented graph $\vec{G}$ with $\operatorname{th}(\vec{G}) \leq 4$ is shown on left and an induced subgraph $\vec{H}$ of $\vec{G}$ is shown on the right with $\operatorname{th}(\vec{H}) = 5$.
  • Figure 2: An undirected graph $G$ with $\operatorname{th}(G) \leq 4$ is shown on left and an induced subgraph $H$ of $G$ is shown on the right with $\operatorname{th}(H) = 5$.
  • Figure 3: The digraph (above) has the following extension (below) by Definition \ref{['extensiondef']}.
  • Figure 4: The graph $H_{3,3}$ is shown.
  • Figure 5: For the graph $\Gamma$ (above), $B$ is the set of blue vertices satisfying $\operatorname{pt}(\Gamma;B)=2$. For $\Gamma^T$ (below), $\operatorname{pt}(\Gamma^T;\operatorname{Term}(\mathcal{F}))=1$, where $\mathcal{F}$ is the unique set of forces of $B$ in $\Gamma$.
  • ...and 5 more figures

Theorems & Definitions (62)

  • Remark 2.1
  • Example 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Theorem 2.6
  • proof
  • Corollary 2.7
  • Proposition 2.8
  • proof
  • ...and 52 more