# 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.

## Key words

Analysis of algorithms*P*-completeness

*NC*algorithms the lexicographically first maximal independent set problem the lexicographically first maximal subgraph problems

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