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Parallel algorithms for finding the connected components (CC) and a minimum spanning FOREST (MSF) of an undirected graph are presented. The primary model of computation considered is that called "shuffle-exchange network" in which each processor has its own local memory, no memory is shared, and communication among processors is done via a fixed degree network. This model is very convenient for actual realization. Both algorithms have depth of O(log2 n) while using n2 processors. Here n is the number of vertices in the graph. The algorithms are first presented for the PRAM (parallel RAM) model, which is not realizable, but much more convenient for the design and presentation of algorithms. The CC and MSF algorithms are no exceptions. The CC PRAM algorithm is a simplification of the one appearing in . A modification of this algorithm yields a simple and efficient MSF algorithm. Both have depth of O(log m) and they use m processors, where m is the number of edges in the graph.