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Chasing puppies on orthogonal straight-line plane graphs

Johanna Ockenfels, Yoshio Okamoto, Patrick Schnider

TL;DR

This work proves that any finite connected graph with an orthogonal straight-line embedding $\gamma$ admits a deterministic strategy for a human to catch a locally optimizing puppy. The authors introduce a recursive, region-forbidding approach that reduces the non-forbidden portion of the embedding, handling both generic cases (no two topmost horizontal edges share height) and non-generic cases (multiple topmost edges) by component domination and a path-structured reduction $G_{\mathcal{D}}$. The method yields a constructive strategy and extends to orthogonal drawings via subdivision, linking to beacon routing and related pursuit-evasion problems. The results advance understanding of pursuit dynamics in orthogonal graph layouts and provide a practical strategy framework for such geometric settings.

Abstract

Assume that you have lost your puppy on an embedded graph. You can walk around on the graph and the puppy will run towards you at infinite speed, always locally minimizing the distance to your current position. Is it always possible for you to reunite with the puppy? We show that if the embedded graph is an orthogonal straight-line embedding the answer is yes.

Chasing puppies on orthogonal straight-line plane graphs

TL;DR

This work proves that any finite connected graph with an orthogonal straight-line embedding admits a deterministic strategy for a human to catch a locally optimizing puppy. The authors introduce a recursive, region-forbidding approach that reduces the non-forbidden portion of the embedding, handling both generic cases (no two topmost horizontal edges share height) and non-generic cases (multiple topmost edges) by component domination and a path-structured reduction . The method yields a constructive strategy and extends to orthogonal drawings via subdivision, linking to beacon routing and related pursuit-evasion problems. The results advance understanding of pursuit dynamics in orthogonal graph layouts and provide a practical strategy framework for such geometric settings.

Abstract

Assume that you have lost your puppy on an embedded graph. You can walk around on the graph and the puppy will run towards you at infinite speed, always locally minimizing the distance to your current position. Is it always possible for you to reunite with the puppy? We show that if the embedded graph is an orthogonal straight-line embedding the answer is yes.

Paper Structure

This paper contains 6 sections, 3 theorems, 6 figures.

Table of Contents

  1. Introduction
  2. Our results
  3. Related work
  4. A Strategy for Orthogonal Embeddings
  5. The generic case
  6. Allowing horizontal edges of the same height

Key Result

Theorem 1.1

Let $G$ be a finite connected graph and let $\gamma$ be an orthogonal straight-line embedding of $G$ in the plane. Then, there is a strategy for the human to catch the puppy on $\gamma$.

Figures (6)

  • Figure 1: A stable configuration (left), and an unstable one (right).
  • Figure 2: If the human $h$ continues walking clockwise, human and puppy $p$ will never be reunited.
  • Figure 3: No matter how the human $h$ moves, human and puppy $p$ will never be reunited.
  • Figure 4: A straight-line embedding (left) and an orthogonal straight-line embedding (right) of the same graph.
  • Figure 5: An example of Step 2 of the algorithm.
  • ...and 1 more figures

Theorems & Definitions (7)

  • Theorem 1.1
  • Conjecture 1.2
  • Lemma 2.1
  • proof
  • proof : Proof of Theorem \ref{['thm:orthogonal_puppies']}
  • Lemma 2.2
  • proof