Another Measure for the Lexicographically First Maximal Subgraph Problems and Its Threshold Value on a Random Graph

We investigate structural parameters of a graph that express the boundary of the parallel complexity of the lexicographically first maximal independent set (LFMIS) problem. The longest directed path length (LDPL), which is introduced in the paper, is a better measure than previously known one. The parallel complexity of the LFMIS problem on a graph gradually increases as the value measured on the graph grows: The LFMIS problem on a graph of LDPL O(log k n) is in NC k+1 , and the problem on a graph of LDPL 2(n ffl ) is P-complete. We also investigate the limit of the measure. We construct a kind of the lexicographically first maximal subgraph problems such that the problem is P-complete even if the LDPL of the input graph is restricted to 1. Finally we discuss the probability that a random graph has LDPL l and show that its threshold value is l n . This implies that a random graph of which each edge exists with probability p has LDPL 2(np) with high probability. The result also ..