Finding a princess in a palace: A pursuit-evasion problem

TL;DR

This paper analyzes a pursuit-evasion game on graphs where a prince seeks a moving princess without knowing her location, characterizing winning graphs and the minimum days needed for the prince to guarantee finding her.

Contribution

It introduces a new pursuit-evasion problem with unknown princess location, characterizes winning graphs, and determines the minimum search time for the prince.

Findings

Characterized graphs where the prince can guarantee to find the princess.

Determined the minimum number of days needed for guaranteed discovery.

Provided strategies for the prince in winning scenarios.

Abstract

This paper solves a pursuit-evasion problem in which a prince must find a princess who is constrained to move on each day from one vertex of a finite graph to another. Unlike the related and much studied `Cops and Robbers Game', the prince has no knowledge of the position of the princess; he may, however, visit any single room he wishes on each day. We characterize the graphs for which the prince has a winning strategy, and determine, for each such graph, the minimum number of days the prince requires to guarantee to find the princess.