Abstract : The surveillance game [Fomin et al., 2012] models the prob- lem of web-page prefetching as a pursuit evasion game played on a graph. This two-player game is played turn-by-turn. The rst player, called the observer, can mark a xed amount of vertices at each turn. The second one controls a surfer that stands at vertices of the graph and can slide along edges. The surfer starts at some initially marked vertex of the graph, her objective is to reach an unmarked node The surveillance number sn(G) of a graph G is the minimum amount of nodes that the observer has to mark at each turn ensuring it wins against any surfer in G. Fomin et al. also de ned the connected surveillance game where the marked nodes must always induce a connected subgraph. They ask if there is a constant c > 0 such that csn(G)/ sn(G) <= c for any graph G. It has been shown that there are graphs G for which csn(G) = sn(G) + 1. In this paper, we investigate this question. We present a family of graphs G such that csn(G) > sn(G)+1. Moreover, we prove that csn(G) <= sn(G) n^(1/2) for any n-node graph G. While the gap between these bounds remains huge, it seems di cult to reduce it. We then de ne the online surveillance game where the observer has no a priori knowledge of the graph topology and discovers it little-by- little. Unfortunately, we show that no algorithm for solving the online surveillance game has competitive ratio better than Omega(Delta).