Mobile robots, especially unmanned aerial vehicles (UAVs), are of increasing interest for surveillance and disaster response scenarios. We consider the problem of multi-robot persistent surveillance with connectivity constraints where robots have to visit sensing locations periodically and maintain a multi-hop connection to a base station. We formally define several problem instances closely related to multi-robot persistent surveillance with connectivity constraints, i.e., connectivity-constrained multi-robot persistent surveillance (CMPS), connectivity-constrained multi-robot reachability (CMR), and connectivity-constrained multi-robot reachability with relay dropping (CMRD), and show that they are all NP-hard on general graph. We introduce three heuristics with different planning horizons for convex grid graphs and combine these with a tree traversal approach which can be applied to a partitioning of non-convex grid graphs (CMPS with tree traversal, CMPSTT). In simulation studies we show that a short horizon greedy approach, which requires parameters to be optimized beforehand, can outperform a full horizon approach, which requires a tour through all sensing locations, if the number of robots is larger than the minimum number of robots required to reach all sensing locations. The minimum number required is the number of robots necessary for building a chain to the farthest sensing location from the base station. Furthermore, we show that partitioning the area and applying the tree traversal approach can achieve a performance similar to the unpartitioned case up to a certain number of robots but requires less optimization time.