Black Hole Search in Directed Graphs
We consider the problem of cooperative network exploration by agents under the assumption that there is a harmful host present in the network that destroys the incoming agents without outside trace – the so-called black hole search problem.
Many variants of this problem have been studied, with various assumptions about the timing, agents’ knowledge about the topology, means of inter-agent communication, amount of writable memory in vertices, and other parameters. However, all this research considered undirected graphs only, and relied to some extent on the ability of an agent to mark an edge as safe immediately after having traversed it.
In this paper we study directed graphs where this technique does not apply, and show that the consequence is an exponential gap: While in undirected graphs Δ + 1 agents are always sufficient, in the directed case at least 2Δ agents are needed in the worst case, where Δ is in-degree of the black hole. This lower bound holds also in the case of synchronous agents. Furthermore, we ask the question What structural information is sufficient to close this gap? and show that in planar graphs with a planar embedding known to the agents, 2Δ + 1 agents are sufficient, and 2Δ agents are necessary.
KeywordsBlack Hole Directed Graph Planar Graph Mobile Agent Safe Area
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