Vince and Wang [6] showed that the average subtree density of a seriesreduced tree is between 1 2 and 3 4 , answering a conjecture of Jamison [4]. They ask under what conditions a sequence of such trees may have average subtree density tending to either bound; we answer these questions by giving simple necessary and sufficient conditions in each case.

We introduce a model of a preferential attachment based random graph which extends the family of models in which condensation phenomena can occur. Each vertex has an associated uniform random variable which we call its location. Our model evolves in discrete time by selecting r vertices from the graph with replacement, with probabilities proportional to their degrees plus a constant α. A new vertex joins the network and attaches to one of these vertices according to a given probability associated to the ranking of their locations. We give conditions for the occurrence of condensation, showing the existence of phase transitions in α below which condensation occurs. The condensation in our model differs from that in preferential attachment models with fitness in that the condensation can occur at a random location, that it can be due to a persistent hub, and that there can be more than one point of condensation.

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)).