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.

翻译：我们研究了一个名为$mathsf{MSO}mathsf{mathsf{GL ⁇ mathsf{Lin}$(美元)的模型检查问题,这是最近由Knop, Kouteck\{y}, Masa\v{r{{{{i}k和Toufar[Log.方法计算.Sci., 15(4), 2019]介绍的,它与当地和全球的主要限制因素有关,称为$mathsfsf{MSO ⁇ mathsf{GL ⁇ mathsf{Lin}$(美元),它叫做$mathsf{MSO{mathsf{GL ⁇ mathsf{Lin{Lin}$(美元),最近由Knop, Koutecklex\\\{{y}, Kousa{{{y}, Masa{r{r{{{{{{{{{{{{{i}k}k, 和Toufar[Logroductroduction comm comm commcut.Supducul.Scidud.Sci.Sci.15(4), 15/2019]。我们表明, 问题是固定的参数可移动参数。我们证明 问题。我们显示, 问题是由脊椎完整性的, 和垂直是固定可移动参数。我们发现, oxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx