Can Romeo and Juliet Meet? Or Rendezvous Games with Adversaries on Graphs

We introduce the rendezvous game with adversaries. In this game, two players, {\sl Facilitator} and {\sl Disruptor}, play against each other on a graph. Facilitator has two agents, and Disruptor has a team of $k$ agents located in some vertices of the graph. They take turns in moving their agents to adjacent vertices (or staying). Facilitator wins if his agents meet in some vertex of the graph. The goal of Disruptor is to prevent the rendezvous of Facilitator's agents. Our interest is to decide whether Facilitator can win. It appears that, in general, the problem is PSPACE-hard and, when parameterized by $k$, co-W[2]-hard. Moreover, even the game's variant where we ask whether Facilitator can ensure the meeting of his agents within $\tau$ steps is co-NP-complete already for $\tau=2$. On the other hand, for chordal and $P_5$-free graphs, we prove that the problem is solvable in polynomial time. These algorithms exploit an interesting relation of the game and minimum vertex cuts in certain graph classes. Finally, we show that the problem is fixed-parameter tractable parameterized by both the graph's neighborhood diversity and $\tau$.