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)).

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 .

is an expertly guided tour of combinatorial optimization. The theme of the tour is the traveling salesman problem (TSP). In the course of exploring different facets of TSP research and results, our guides expose concepts, methods, and ideas from computational complexity, graph theory, probabilistic analysis of algorithms, statistical methods for comparing heuristics, polyhedral theory, and more generally, the entire field of combinatorial optimization. This well-written book more than fulfills the editors' hopes of providing comprehensive coverage of the techniques of combinatorial optimization as well as being a state-of-the-art survey of the traveling salesman problem. We highly recommend this book to advanced students and researchers in operations research, computer science, discrete mathematics, and quantitative business studies.The TSP is a classic 'hard' problem: easy to describe and difficult to solve. Precisely how difficult became clearer as computational complexity theory developed. Chapter 1 (A.J. Hoffman and P. Wolfe) explores the history of the problem. In Chapter 2, R.S. Garfinkel discusses motivations for the problem and some of its many applications and generalizations. Chapter 3 (D.S. Johnson and C.H. Papadimitriou) introduces computational complexity theory and shows how the difficulty of the TSP optimization problem can be specified as NP-hard. This chapter delivers the bad news of the TSP's intractability. P.C. Gilmore, E.L. Lawler, and D.B. Shmoys survey well-solved special cases of the TSP in Chapter 4. Several results are new. The authors categorize these special cases (those with restrictions on arc lengths and those with network structures) and show how the theory of subtour patching is used to give optimal solutions to several special cases. Knowing the well-solved

is an expertly guided tour of combinatorial optimization. The theme of the tour is the traveling salesman problem (TSP). In the course of exploring different facets of TSP research and results, our guides expose concepts, methods, and ideas from computational complexity, graph theory, probabilistic analysis of algorithms, statistical methods for comparing heuristics, polyhedral theory, and more generally, the entire field of combinatorial optimization. This well-written book more than fulfills the editors' hopes of providing comprehensive coverage of the techniques of combinatorial optimization as well as being a state-of-the-art survey of the traveling salesman problem. We highly recommend this book to advanced students and re-

We consider a game in which a cop searches for a moving robber on a graph using distance probes, which is a slight variation on one introduced by Seager. 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. They conjectured that this bound was best possible for complete graphs, but the present authors showed that in fact the cop wins on K 1/m n if and only if m n/2, for all but a few small values of n. In this paper we extend this result to general graphs by proving that the cop has a winning strategy on G 1/m provided m n/2 for all but a few small values of n; this bound is best possible. We also consider replacing the edges of G with paths of varying lengths.