# The lexicographically first maximal subgraph problems:*P*-completeness and*NC* algorithms

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## Abstract

The lexicographically first maximal (lfm) subgraph problem for a property π is to compute the lfm vertex set whose induced subgraph satisfies π. The main contribution of this paper is the*P*-completeness of the lfm subgraph problem for any nontrivial hereditary property. We also observe that most of the lfm subgraph problems are still*P*-complete even if the instances are restricted to graphs with degree 3. However, some exceptions are found. For example, it is shown that the lfm 4-cycle free subgraph problem is in*NC*^{2} for graphs with degree 3 but turns out to be*P*-complete for graphs with degree 4. Further, we analyze the complexity of the lfm*edge-induced* subgraph problem for some graph properties and show that it has a different complexity feature.

## Keywords

Computational Mathematic Complexity Feature Graph Property Hereditary Property Subgraph Problem## Preview

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