A measure of parallelization for the lexicographically first maximal subgraph problems

Abstract

A maximum directed tree size (MDTS) is defined by the maximum number of the vertices of a directed tree on the directed acyclic graph of a given undirected graph. The MDTS of a graph measures the parallelization for the lexicographically first maximal subgraph (LFMS) problems. That is, the complexity of the problems on a graph family $\mathcal{G}$gradually increases as the value measured on each graph in the family grows; (1) if the MDTS of each graph in $\mathcal{G}$is O(log^k n), the lexicographically first maximal independent set problem on $\mathcal{G}$is in NC ^k+1 and the LFMS problem for π is in NC ^k+S, where π is a property on graphs such that π is nontrivial, hereditary, and NC ^s−1 testable; (2) both problems above are P-complete if the MDTS of each graph in $\mathcal{G}$is cn ^e. It is worth remarking that the problem to compute the MDTS is in NC ² this is important in the sense that a “measure” means only if measuring the complexity of a problem is easier than solving the problem.