Extended MSO Model Checking via Small Vertex Integrity

We study the model checking problem of an extended mathsfMSO with local and global cardinality constraints, called mathsfMSO^mathsfGL_mathsfLin, introduced recently by Knop, Kouteck'y, Masavr'ik, 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.