Search

We consider the Cops and Robber pursuit-evasion game when the edge-set of the graph is allowed to change in time, possibly at every round. Specifically, the game is played on an infinite periodic sequence $\mathcal{G} = (G_0, \dots, G_{p-1})^*$ of graphs on the same set $V$ of $n$ vertices: in round $t$, the topology of $\mathcal{G}$ is $G_i=(V,E_i)$ where $i\equiv t \pmod{p}$. Concentrating on the case of a single cop, we provide a characterization of copwin periodic temporal graphs, establishing several basic properties on their nature, and extending to the temporal domain classical C&R concepts such as covers and corners. Based on these results, we design an efficient algorithm for determining if a periodic temporal graph is copwin. We also consider the case of $k>1$ cops. By shifting from a representation in terms of directed graphs to one in terms of directed multi-hypergraphs, we prove that all the fundamental properties established for $k=1$ continue to hold, providing a characterization of $k$-copwin periodic graphs, as well as a general strategy to determine if a periodic graph is $k$-copwin. Our results do not rely on any assumption on properties such as connectivity, symmetry, reflexivity held by the individual graphs in the sequence. They are established for a unified version of the game that includes the standard games studied in the literature, both for undirected and directed graphs, and both when the players are fully active and when they are not. They hold also for a variety of settings not considered in the literature.

In this paper we investigate the problem of searching for a black hole in a dynamic graph by a set of scattered agents (i.e., the agents start from arbitrary locations of the graph). The black hole is a node that silently destroys any agent visiting it. This kind of malicious node nicely models network failures such as a crashed host or a virus that erases the visiting agents. The black hole search problem is solved when at least one agent survives, and it has the entire map of the graph with the location of the black hole. We consider the case in which the underlining graph is a dynamic 1-interval connected ring: a ring graph in which at each round at most one edge can be missing. We first show that the problem cannot be solved if the agents can only communicate by using a face-to-face mechanism: this holds for any set of agents of constant size, with respect to the size $n$ of the ring. To circumvent this impossibility we consider agents equipped with movable pebbles that can be left on nodes as a form of communication with other agents. When pebbles are available, three agents can localize the black hole in $O(n^2)$ moves. We show that such a number of agents is optimal. We also show that the complexity is tight, that is $\Omega(n^2)$ moves are required for any algorithm solving the problem with three agents, even with stronger communication mechanisms (e.g., a whiteboard on each node on which agents can write messages of unlimited size). To the best of our knowledge this is the first paper examining the problem of searching a black hole in a dynamic environment with scattered agents., Comment: arXiv admin note: text overlap with arXiv:2005.07453