Extended MSO Model Checking via Small Vertex Integrity
We study the model checking problem of an extended ๐ฌ๐ฒ๐ฎ with local and global cardinality constraints, called ๐ฌ๐ฒ๐ฎ^๐ฆ๐ซ_๐ซ๐๐, introduced recently by Knop, Kouteckรฝ, Masaลรญk, 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 fill a gap between the fixed-parameter tractability parameterized by vertex cover number and the W[1]-hardness parameterized by treedepth.
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