Exploring the Gap Between Treedepth and Vertex Cover Through Vertex Integrity

Extended MSO Model Checking via Small Vertex Integrity

“…We show that this problem, called MSO GL Lin Model Checking, is fixed-parameter tractable parameterized by vertex integrity. This result fills a missing part in the map on the complexity of MSO GL Lin Model Checking as vertex integrity fits between these parameters [35,45] (see Figure 1). Note that by MSO, we mean MSO 1 .…”

“…The fracture number was used to design efficient algorithms for Integer Linear Programming [19], Bounded-Degree Vertex Deletion [33], and Locally Constrained Homomorphism [9]. The vertex integrity was used in the context of subgraph isomorphism on minor-closed graph classes [7], and then used to design algorithm for several problems that are hard on graphs of bounded treedepth such as Capacitated Dominating Set, Capacitated Vertex Cover, Equitable Coloring, Equitable Connected Partition 3 , Imbalance, Maximum Common (Induced) Subgraph, and Precoloring Extension [35]. A faster algorithm for MSO Model Checking parameterized by vertex integrity is also known [45].…”

“…Indeed, this problem is known to be W [1]-hard parameterized by r and treewidth [21]. 1 See [3,35,49] for many other examples of such problems.…”

“…For example, by techniques tailored for individual problems, several problems are shown to be tractable on graphs of bounded vertex cover number (see e.g., [20,22,24]). Such results are known also for more general parameters such as twin-cover [31], neighborhood diversity [44], and vertex integrity [35]. Then the natural challenge would be finding a meta-theorem on those smaller graph classes but for a wider variety of target problems.…”

“…On the other hand, the concept of treedepth can be seen as a recursive generalization of vertex integrity. Actually, their definitions give us the inequality treedepth ≤ vertex integrity ≤ vertex cover number − 1 for every graph (see [35]).…”