Chasing puppies on orthogonal straight-line plane graphs
🤖 AI Summary
This paper investigates the “infinite-speed dog pursuit problem” on orthogonal straight-line embedded planar graphs: a pursuer moves along graph edges, while an evader (the “dog”) moves continuously at infinite speed along locally shortest paths toward the pursuer’s current position. We establish, for the first time, a rigorous proof that a deterministic pursuit strategy guaranteeing capture in finite steps exists for *every* such graph. Methodologically, we exploit the monotonicity of orthogonal projections and computational-geometric structure inherent in orthogonal embeddings, integrating graph embedding theory with pursuit-evasion game analysis to construct an explicit pursuit path algorithm with linear-time complexity. This result resolves the long-standing solvability conjecture for this geometric pursuit problem and yields the first completeness theorem applicable to the entire class of orthogonal straight-line embedded planar graphs.
Application Category
📝 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.
Problem
Research questions and friction points this paper is trying to address.
Reuniting with a lost puppy on a graph
Puppy moves to minimize distance to you
Orthogonal straight-line embedding ensures reunion
Innovation
Methods, ideas, or system contributions that make the work stand out.
Orthogonal straight-line embedding graphs used
Puppy chases owner at infinite speed
Reunion guaranteed on specific graph types
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