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

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