Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Deduction with $k$ moves

Andrea C. Burgess, Nancy E. Clarke, Shannon L. Fitzpatrick, Melissa A. Huggan

TL;DR

This work extends the deduction game to allow each searcher to move up to $k$ times, introducing the $k$-move deduction number $d_k(G)$. It develops general bounds and exact values for basic graphs (e.g., $d_k(P_n)=\left\lceil n/(k+1)\right\rceil$, $d_k(C_n)=\max\{2,\left\lceil n/(k+1)\right\rceil\}$, $d_k(K_n)=n-1$) and analyzes $d_k$ on Cartesian and strong products, deriving lower bounds and many exact results for products of paths. The paper provides constructive layouts yielding tight bounds in many regimes, illustrating how product structure influences multi-move search strategies. These results advance understanding of multi-stage, locally coordinated search on graphs and have potential implications for distributed surveillance and planning problems where communication is restricted.

Abstract

The deduction game may be thought of as a variant on the classical game of cops and robber in which the cops (searchers) aim to capture an invisible robber (evader); each cop is allowed to move at most once, and cops situated on different vertices cannot communicate to co-ordinate their strategy. In this paper, we extend the deduction game to allow each searcher to make $k$ moves, where $k$ is a fixed positive integer. We consider the value of the $k$-move deduction number on several classes of graphs including paths, cycles, complete graphs, complete bipartite graphs, and Cartesian and strong products of paths.

Deduction with $k$ moves

TL;DR

This work extends the deduction game to allow each searcher to move up to times, introducing the -move deduction number . It develops general bounds and exact values for basic graphs (e.g., , , ) and analyzes on Cartesian and strong products, deriving lower bounds and many exact results for products of paths. The paper provides constructive layouts yielding tight bounds in many regimes, illustrating how product structure influences multi-move search strategies. These results advance understanding of multi-stage, locally coordinated search on graphs and have potential implications for distributed surveillance and planning problems where communication is restricted.

Abstract

The deduction game may be thought of as a variant on the classical game of cops and robber in which the cops (searchers) aim to capture an invisible robber (evader); each cop is allowed to move at most once, and cops situated on different vertices cannot communicate to co-ordinate their strategy. In this paper, we extend the deduction game to allow each searcher to make moves, where is a fixed positive integer. We consider the value of the -move deduction number on several classes of graphs including paths, cycles, complete graphs, complete bipartite graphs, and Cartesian and strong products of paths.

Paper Structure

This paper contains 12 sections, 30 theorems, 18 equations, 8 figures.

Table of Contents

  1. Introduction
  2. $k$-move deduction as a search model
  3. Initial Results
  4. Cartesian Products
  5. Upper and Lower Bounds on $d_k(G \Box H)$
  6. Cartesian Products of Paths
  7. Strong Products
  8. Lower Bounds on $d_k(G\boxtimes H)$
  9. Strong Product of Paths
  10. Discussion
  11. Statements and Declarations
  12. Acknowledgments

Key Result

Theorem 1.1

In any connected graph $G$ of order $n \geq 3$ which contains $\ell$ vertices of degree $1$, where $\alpha(G)$ is the independence number of $G$ and $\omega(G)$ is the clique number of $G$.

Figures (8)

  • Figure 1: $P_m \Box P_n$ can be partitioned into two subgrids $P_m \Box P_{n_1}$ and $H_2\cong P_m \Box P_{n_2}$
  • Figure 2: Successful initial layout for $P_5 \Box P_5$ when $k=2$
  • Figure 3: An illustration of the searcher movements in the proof of Lemma \ref{['lem:halfway1']}
  • Figure 4: An illustration of the searcher movements in the proof of Lemma \ref{['lem: multi upper bound']}
  • Figure 5: Two of the starting configurations for searchers in the subgraph $H$ pictured in Figure \ref{['thirdL']}
  • ...and 3 more figures

Theorems & Definitions (57)

  • Theorem 1.1: BDF2024BDOWXY
  • Lemma 2.1
  • proof
  • Corollary 2.2
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • ...and 47 more