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 $\tau$.

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罗密欧与朱丽叶可以见面吗？或在图形上带有对手的交会游戏

我们介绍了与敌人会合的游戏。在此游戏中，两个玩家{\ sl Facilitator}和{\ sl Disruptor}在图表上相互对战。主持人有两个业务代表，Disruptor在图形的某些顶点上有一个由$ k $的业务代表组成的团队。他们轮流将他们的代理移动到相邻的顶点（或停留）。如果其代理人在图表的某个顶点相遇，则主持人会获胜。Disruptor的目标是防止协调员的代理。我们的兴趣是确定协调人是否可以获胜。看来，一般来说，问题是PSPACE困难的，当用$ k $参数化时，问题是co-W [2]困难的。而且，即使是游戏的变体，我们询问协调人是否可以确保在$ \ tau $步骤之内与他的代理人会面，对于$ \ tau = 2 $，这已经是共NP完成的。另一方面，对于和弦和无$ P_5 $的图，我们证明问题可以在多项式时间内解决。这些算法利用了某些图形类中游戏与最小顶点割线之间的有趣关系。最后，我们证明了该问题是由图的邻域多样性和$ \ tau $均参数化的固定参数易处理参数。