Extended MSO Model Checking via Small Vertex Integrity

TL;DR

This paper proves that model checking for an extended monadic second-order logic with cardinality constraints is fixed-parameter tractable when parameterized by vertex integrity, a graph measure between vertex cover and treedepth.

Contribution

It establishes fixed-parameter tractability of the extended MSO model checking problem with respect to vertex integrity, bridging the gap between known tractable and hard cases.

Findings

Model checking is fixed-parameter tractable with vertex integrity.

Vertex integrity is a key parameter between vertex cover and treedepth.

Results narrow the gap between tractability and hardness in graph parameters.

Abstract

We study the model checking problem of an extended $\mathsf{MSO}$ with local and global cardinality constraints, called $\mathsf{MSO}^{\mathsf{GL}}_{\mathsf{Lin}}$, introduced recently by Knop, Kouteck\'{y}, Masa\v{r}\'{i}k, and Toufar [Log. Methods Comput. Sci., 15(4), 2019]. We show that the problem is fixed-parameter tractable parameterized by vertex integrity, where vertex integrity is a graph parameter standing between vertex cover number and treedepth. Our result thus narrows the gap between the fixed-parameter tractability parameterized by vertex cover number and the W[1]-hardness parameterized by treedepth.