Abstract
In this paper, we study the necessary and sufficient time to explore constantly connected dynamics graphs by a mobile entity (agent). A dynamic graph is constantly connected if for each time units, there exists a stable connected spanning tree [10]. We focus on the case where the underlying graph is a cactus-path (graph reduced to a path of k rings in which two neighbor rings have at most one vertex in common) and we assume that the agent knows the dynamics of the graph. We show that 5n - Θ(1) time units are necessary and sufficient to explore any constantly connected dynamic graph based on the cactus-path 〖Ch〗_(2,n) (composed of two same size ringsn). The upper bound is generalized on dynamic graphs based on cacti-paths with k rings. We show that for any constantly connected dynamic graph of size N based on a cactus-path, 4N -max{n_1,n_k} -3k -3 time units are sufficient to explore the graph, with k the length of the path, N=∑_(i=1)^k▒n_i -k+1 the size of the dynamic graph and n_i the size of the ring which is at position i starting from left to right.
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