Can Romeo and Juliet meet? Or rendezvous games with adversaries on graphs

Abstract

We introduce the rendezvous game with adversaries. In this game, two players, Facilitator and Divider, play against each other on a graph. Facilitator has two agents and Divider has a team of k agents located in some vertices. They take turns in moving their agents to adjacent vertices (or staying put). Facilitator wins if his agents meet in some vertex. Divider aims to prevent the rendezvous of Facilitator's agents. We show that deciding whether Facilitator can win is PSPACE-hard and, when parameterized by k, co-W[2]-hard. Moreover, even deciding whether Facilitator can win within τ steps is co-NP-complete already for τ=2. On the other hand, for chordal and P5-free graphs, we prove that the problem is solvable in polynomial time. Finally, we show that the problem is fixed-parameter tractable parameterized by both the graph's neighborhood diversity and the number of steps τ.