In this paper, we investigate the following question: "How can a group of initially co-located mobile agents perpetually explore an unknown graph, when one stationary node occasionally behaves maliciously, under the control of an adversary?" This malicious node is termed as "Byzantine black hole (BBH)" and at any given round it may choose to destroy all visiting agents, or none of them. While investigating this question, we found out that this subtle power turns out to drastically undermine even basic exploration strategies which have been proposed in the context of a classical, always active, black hole. We study this perpetual exploration problem in the presence of at most one BBH, without initial knowledge of the network size. Since the underlying graph may be 1-connected, perpetual exploration of the entire graph may be infeasible. Accordingly, we define two variants of the problem, termed as PerpExploration-BBH and PerpExploration-BBH-Home. In the former, the agents are tasked to perform perpetual exploration of at least one component, obtained after the exclusion of the BBH. In the latter, the agents are tasked to perform perpetual exploration of the component which contains the home node, where agents are initially co-located. Naturally, PerpExploration-BBH-Home is a special case of PerpExploration-BBH. The mobile agents are controlled by a synchronous scheduler, and they communicate via face-to-face model of communication. The main objective in this paper is to determine the minimum number of agents necessary and sufficient to solve these problems. We first consider the problems in acyclic networks, and we obtain optimal algorithms that solve PerpExploration-BBH with 4 agents, and PerpExploration-BBH-Home with 6 agents in trees. The lower bounds hold even in path graphs. In general graphs, we give a non-trivial lower bound of 2Δ-1 agents for PerpExploration-BBH, and an upper bound of 3Δ+3 agents for PerpExploration-BBH-Home. To the best of our knowledge, this is the first paper that studies a variant of a black hole in arbitrary networks, without initial topological knowledge about the network.

Perpetual exploration is a fundamental problem in the domain of mobile agents, where an agent needs to visit each node infinitely often. This issue has received lot of attention, mainly for ring topologies, presence of black holes adds more complexity. A black hole can destroy any incoming agent without any observable trace. In \cite{BampasImprovedPeriodicDataRetrieval,KralovivcPeriodicDataRetrievalFirst}, the authors considered this problem in the context of \textit{ Periodic data retrieval}. They introduced a variant of black hole called gray hole (where the adversary chooses whether to destroy an agent or let it pass) among others and showed that 4 asynchronous and co-located agents are essential to solve this problem (hence perpetual exploration) in presence of such a gray hole if each node of the ring has a whiteboard. This paper investigates the exploration of a ring in presence of a ``byzantine black hole''. In addition to the capabilities of a gray hole, in this variant, the adversary chooses whether to erase any previously stored information on that node. Previously, one particular initial scenario (i.e., agents are co-located) and one particular communication model (i.e., whiteboard) are investigated. Now, there can be other initial scenarios where all agents may not be co-located. Also, there are many weaker models of communications (i.e., Face-to-Face, Pebble) where this problem is yet to be investigated. The agents are synchronous. The main results focus on minimizing the agent number while ensuring that perpetual exploration is achieved even in presence of such a node under various communication models and starting positions. Further, we achieved a better upper and lower bound result (i.e., 3 agents) for this problem (where the malicious node is a generalized version of a gray hole), by trading-off scheduler capability, for co-located and in presence of a whiteboard.