FO Model Checking of Interval Graphs

TL;DR

This paper investigates the computational complexity of first-order (FO) model checking on interval graphs, showing efficient algorithms for certain interval length restrictions and hardness results for others.

Contribution

It provides a new efficient algorithm for FO model checking on interval graphs with finite interval length sets and proves hardness for dense length sets.

Findings

FO model checking is solvable in O(n log n) for interval graphs with finite interval lengths.

The same problem is not efficiently solvable when interval lengths are from dense sets.

Efficient algorithms exist for specific restricted classes of interval graphs.

Abstract

We study the computational complexity of the FO model checking problem on interval graphs, i.e., intersection graphs of intervals on the real line. The main positive result is that FO model checking and successor-invariant FO model checking can be solved in time O(n log n) for n-vertex interval graphs with representations containing only intervals with lengths from a prescribed finite set. We complement this result by showing that the same is not true if the lengths are restricted to any set that is dense in an open subset, e.g., in the set $(1, 1 + \varepsilon)$.