For intractable problems on graphs of bounded treewidth, two graph parameters treedepth and vertex cover number have been used to obtain fine-grained complexity results. Although the studies in this direction are successful, we still need a systematic way for further investigations because the graphs of bounded vertex cover number form a rather small subclass of the graphs of bounded treedepth. To fill this gap, we use vertex integrity, which is placed between the two parameters mentioned above. For several graph problems, we generalize fixed-parameter tractability results parameterized by vertex cover number to the ones parameterized by vertex integrity. We also show some finer complexity contrasts by showing hardness with respect to vertex integrity or treedepth.

翻译：对于树宽受限的图上的难解问题，已使用树深度和顶点覆盖数这两个图参数来获得细粒度复杂性结果。尽管这一方向的研究取得了成功，但由于顶点覆盖数有界的图构成树深度有界图的一个相当小的子类，我们仍需一种系统性的方法进行进一步探索。为填补这一空白，我们采用顶点完整性这一参数，该参数介于上述两个参数之间。对于多个图问题，我们将参数化为顶点覆盖数的固定参数可处理性结果推广到参数化为顶点完整性的情形。通过展示关于顶点完整性或树深度的难解性，我们还给出了一些更精细的复杂性对比。