Exploring the Gap Between Treedepth and Vertex Cover Through Vertex Integrity

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.