We consider a game in which a cop searches for a moving robber on a connected graph using distance probes, which is a slight variation on one introduced by Seager (Seager, 2012). Carragher, Choi, Delcourt, Erickson and West showed that for any n-vertex graph G there is a winning strategy for the cop on the graph G 1/m obtained by replacing each edge of G by a path of length m, if m ≥ n (Carragher et al., 2012). The present authors showed that, for all but a few small values of n, this bound may be improved to m ≥ n/2, which is best possible (Haslegrave et al., 2016b). In this paper we consider the natural extension in which the cop probes a set of k vertices, rather than a single vertex, at each turn. We consider the relationship between the value of k required to ensure victory on the original graph with the length of subdivisions required to ensure victory with k = 1. We give an asymptotically best-possible linear bound in one direction, but show that in the other direction no subexponential bound holds. We also give a bound on the value of k for which the cop has a winning strategy on any (possibly infinite) connected graph of maximum degree ∆, which is best possible up to a factor of (1 − o(1)).

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We consider a game in which a cop searches for a moving robber on a graph using distance probes, studied by Carragher, Choi, Delcourt, Erickson and West, which is a slight variation on one introduced by Seager. Carragher, Choi, Delcourt, Erickson and West show that for any fixed graph G there is a winning strategy for the cop on the graph G 1/m , obtained by replacing each edge of G by a path of length m, if m is sufficiently large. They conjecture that the cop does not have a winning strategy on K 1/m n if m < n; we show that in fact the cop wins if and only if m n/2, for all but a few small values of n. They also show that the robber can avoid capture on any graph of girth 3, 4 or 5, and ask whether there is any graph of girth 6 on which the cop wins. We show that there is, but that no such graph can be bipartite; in the process we give a counterexample for their conjecture that the set of graphs on which the cop wins is closed under the operation of subdividing edges. We also give a complete answer to the question of when the cop has a winning strategy on K 1/m a,b .

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Graph bootstrap percolation, introduced by Bollobás in 1968, is a cellular automaton defined as follows. Given a "small" graph H and a "large" graph G = G 0 ⊆ K n , in consecutive steps we obtain G t+1 from G t by adding to it all new edges e such that G t ∪ e contains a new copy of H. We say that G percolates if for some t ≥ 0, we have G t = K n .For H = K r , the question about the size of the smallest percolating graphs was independently answered by Alon, Frankl and Kalai in the 1980's. Recently, Balogh, Bollobás and Morris considered graph bootstrap percolation for G = G(n, p) and studied the critical probability p c (n, K r ), for the event that the graph percolates with high probability. In this paper, using the same setup, we determine, up to a logarithmic factor, the critical probability for percolation by time t for all 1 ≤ t ≤ C log log n.The study of critical probabilities has brought numerous and often very sharp results for various graphs G and the values of the infection threshold. For example, van Enter [14] and Schonmann [20] studied r-neighbour bootstrap percolation on Z d , Holroyd [16], Balogh, Bollobás and Morris [4], Balogh, Bollobás, Duminil-Copin and Morris [3] analysed finite grids, while Balogh and Pittel [6], Janson, Łuczak, Turova and Vallier [17] and Bollobás, Gunderson, Holmgren, Janson and Przykucki [10] worked with random graphs.

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Summary In this articles adjacency-preserving mappings from a graph to another are formalized in the Mizar system [7], [2]. The generality of the approach seems to be largely unpreceeded in the literature to the best of the author’s knowledge. However, the most important property defined in the article is that of two graphs being isomorphic, which has been extensively studied. Another graph decorator is introduced as well.

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