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Deduction, Constrained Zero Forcing, and Constrained Searching

Andrea Burgess, Danny Dyer, Kerry Ojakian, Lusheng Wang, Mingyu Xiao, Boting Yang

TL;DR

It is shown that the minimum number of searchers required to clear the graph is the same in deduction as in constrained versions of other previously studied graph processes, namely zero forcing and fast-mixed search.

Abstract

Deduction is a recently introduced graph searching process in which searchers clear the vertex set of a graph with one move each, with each searcher's movement determined by which of its neighbors are protected by other searchers. In this paper, we show that the minimum number of searchers required to clear the graph is the same in deduction as in constrained versions of other previously studied graph processes, namely zero forcing and fast-mixed search. We give a structural characterization, new bounds and a spectrum result on the number of searchers required. We consider the complexity of computing this parameter, giving an NP-completeness result for arbitrary graphs, and exhibiting families of graphs for which the parameter can be computed in polynomial time. We also describe properties of the deduction process related to the timing of searcher movement and the success of terminal layouts.

Deduction, Constrained Zero Forcing, and Constrained Searching

TL;DR

It is shown that the minimum number of searchers required to clear the graph is the same in deduction as in constrained versions of other previously studied graph processes, namely zero forcing and fast-mixed search.

Abstract

Deduction is a recently introduced graph searching process in which searchers clear the vertex set of a graph with one move each, with each searcher's movement determined by which of its neighbors are protected by other searchers. In this paper, we show that the minimum number of searchers required to clear the graph is the same in deduction as in constrained versions of other previously studied graph processes, namely zero forcing and fast-mixed search. We give a structural characterization, new bounds and a spectrum result on the number of searchers required. We consider the complexity of computing this parameter, giving an NP-completeness result for arbitrary graphs, and exhibiting families of graphs for which the parameter can be computed in polynomial time. We also describe properties of the deduction process related to the timing of searcher movement and the success of terminal layouts.

Paper Structure

This paper contains 11 sections, 26 theorems, 8 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Standard layouts and free deduction
  3. Equivalence Theorem
  4. Basic Properties
  5. Dismantlable graphs
  6. Pendent edge-dismantlable
  7. Clique-constructable
  8. Complexity
  9. Algorithms
  10. Algorithms on Cacti
  11. Discussion

Key Result

Theorem 2.1

BDF For the deduction game, if there is a successful layout with $k$ searchers then there is a successful standard layout with $k$ searchers.

Figures (7)

  • Figure 1: A graph $G$ with $\mathbf{czf}(G) = 4$, shown (a) initially; (b) after one vertex is forced; (c) after two vertices are forced; and (d) at the end of the process.
  • Figure 2: A graph $G$ with $\mathbf{cfms}(G) = 4$, shown (a) initially; (b) after sliding from $v_1$ to $v_2$; (c) after sliding from $v_4$ to $v_5$; and (d) after sliding from $v_6$ to $v_7$.
  • Figure 3: A graph $G$ with $\mathbf{d}(G) = 4$, shown (a) initially; (b) after the first stage; (c) after the second stage; and (d) at the end of the process.
  • Figure 4: A graph $G$ with $\mathbf{mu}(G) = 3$, shown with (a) $V(M_1) = V(G) - \{v_3\}$ and (b) $V(M_2) = V(G) - \{v_1\}$.
  • Figure 5: Two more final configurations of deduction searchers in $G$ if (a) the first vertex fired is $v_1$ and (b) the first vertex fired is $v_3$.
  • ...and 2 more figures

Theorems & Definitions (47)

  • Theorem 2.1
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 4.1
  • ...and 37 more