Parallel Complexity of the Lexicographically First Maximal Subgraph Problems on Restricted Graphs

Longest directed path length (LDPL) is de ned by the length of the longest path of the directed acyclic graph of a given undirected graph. The LDPL of a graph measures the parallelization for the lexicographically rst maximal subgraph (LFMS) problems. That is, the parallel complexity of the problems gradually increases as the value measured on a graph grows; the LFMS problems are in NC on graphs of LDPL O(log k n), and P-complete on graphs of LDPL cn . It is worth remarking that the problem to compute the LDPL is in NC 2 . The LDPL on a random graph is also considered, and the threshold value for the LDPL l is l n for a random graph G(n; p).