Fast Collaborative Graph Exploration We study the following scenario of online graph exploration. A team of k agents is initially located at a distinguished vertex r of an undirected graph. In each time step, each agent can traverse an edge of the graph. All vertices have unique identifiers, and upon entering a vertex, an agent obtains a list of the identifiers of all its neighbors. We ask how many time steps are required to complete exploration, i.e., to make sure that every vertex has been visited by some agent. We consider two communication models: one in which all agents have global knowledge of the state of the exploration, and one in which agents may only exchange information when simultaneously located at the same vertex. As our main result, we provide the first strategy which performs exploration of a graph with n vertices at a distance of at most D from r in time O(D), using a team of agents of polynomial size k  D n1 epsilon   0. Our strategy works in the local communication model, without knowledge of global parameters such as n or D. For any constant cu003e1, we show that in the global communication model, a team of k  D nc agents can always complete exploration in D(1 frac1c1 o(1)) time steps, whereas at least D(1 frac1c o(1)) steps are sometimes required. In the local communication model, D(1 frac2c1 o(1)) steps always suffice to complete exploration, and at least D(1 frac2c o(1)) steps are sometimes required.

Cover Time in EdgeUniform StochasticallyEvolving Graphs We define a general model of stochasticallyevolving graphs, namely the edgeuniform stochasticallyevolving graphs. In this model, each possible edge of an underlying general static graph evolves independently being either alive or dead at each discrete time step of evolution following a (Markovian) stochastic rule. The stochastic rule is identical for each possible edge and may depend on the past     k xe2x89xa5 0     observations of the edgexe2x80x99s state. We examine two kinds of random walks for a single agent taking place in such a dynamic graph: (i) The Random Walk with a Delay (RWD), where at each step, the agent chooses (uniformly at random) an incident possible edge, i.e., an incident edge in the underlying static graph, and then, it waits till the edge becomes alive to traverse it. (ii) The more natural Random Walk on what is Available (RWA), where the agent only looks at alive incident edges at each time step and traverses one of them uniformly at random. Our study is on bounding the cover time, i.e., the expected time until each node is visited at least once by the agent. For RWD, we provide a first upper bound for the cases     k  0 , 1     by correlating RWD with a simple random walk on a static graph. Moreover, we present a modified electrical network theory capturing the     k  0     case. For RWA, we derive some first bounds for the case     k  0    , by reducing RWA to an RWDequivalent walk with a modified delay. Further, we also provide a framework that is shown to compute the exact value of the cover time for a general family of stochasticallyevolving graphs in exponential time. Finally, we conduct experiments on the cover time of RWA in edgeuniform graphs and compare the experimental findings with our theoretical bounds.