Deletion to Scattered Graph Classes I -- case of finite number of graph classes

TL;DR

This paper studies a graph modification problem where the goal is to delete a limited number of vertices so that each connected component of the resulting graph belongs to one of a finite set of simpler classes, establishing fixed-parameter tractability under certain conditions.

Contribution

It introduces the problem of deletion to scattered graph classes and proves its fixed-parameter tractability when each class's deletion problem is FPT and class membership is CMSO-expressible.

Findings

The problem is fixed-parameter tractable under specified conditions.

Faster algorithms are developed for classes with finite forbidden sets.

The approach generalizes existing deletion problems to multiple classes.

Abstract

Graph-modification problems, where we modify a graph by adding or deleting vertices or edges or contracting edges to obtain a graph in a {\it simpler} class, is a well-studied optimization problem in all algorithmic paradigms including classical, approximation and parameterized complexity. Specifically, graph-deletion problems, where one needs to delete a small number of vertices to make the resulting graph to belong to a given non-trivial hereditary graph class, captures several well-studied problems including {\sc Vertex Cover}, {\sc Feedback Vertex Set}, {\sc Odd Cycle Transveral}, {\sc Cluster Vertex Deletion}, and {\sc Perfect Deletion}. Investigation into these problems in parameterized complexity has given rise to powerful tools and techniques. We initiate a study of a natural variation of the problem of deletion to {\it scattered graph classes}. We want to delete at most $k$ vertices so that in the resulting graph, each connected component belongs to one of a constant number of graph classes. As our main result, we show that this problem is fixed-parameter tractable (FPT) when the deletion problem corresponding to each of the finite number of graph classes is known to be FPT and the properties that a graph belongs to any of the classes is expressible in Counting Monodic Second Order (CMSO) logic. While this is shown using some black box theorems in parameterized complexity, we give a faster FPT algorithm when each of the graph classes has a finite forbidden set.