Nowadays there exist many networks with dynamic characteristics (e.g. MANETs, DTNs, opportunistic networks, etc.). Our aim is thus to determine the necessary and sufficient connectivity conditions which allow the correct execution of some distributed algorithms on top of them. To this end, we use a framework proposed by Casteigts et al. [1] that exploits the evolving graphs theory, which models the topology of the network as well as the graph relabeling approach, which abstracts communication. In particular, we provide necessary and sufficient conditions for two important distributed problems: mutual exclusion and K-mutual exclusion.