Abstract
Let \(\varPi \) be a hereditary graph class. The problem of deletion to \(\varPi \), takes as input a graph G and asks for a minimum number (or a fixed integer k) of vertices to be deleted from G so that the resulting graph belongs to \(\varPi \). This is a well-studied problem in paradigms including approximation and parameterized complexity. This problem, for example, generalizes vertex cover, feedback vertex set, cluster vertex deletion, perfect deletion to name a few. The study of this problem in parameterized complexity has resulted in several powerful algorithmic techniques including iterative compression and important separators.
Recently, the study of a natural extension of the problem was initiated where we are given a finite set of hereditary graph classes, and the goal is to determine whether k vertices can be deleted from a given graph, so that the connected components of the resulting graph belong to one of the given hereditary graph classes. The problem has been shown to be FPT as long as the deletion problem to each of the given hereditary graph classes is fixed-parameter tractable, and the property of being in any of the graph classes can be expressible in the counting monodic second order (CMSO) logic. While this was shown using some black box theorems, faster algorithms were shown when each of the hereditary graph classes has a finite forbidden set.
In this paper, we do a deep dive on pairs of specific graph classes (\(\varPi _1, \varPi _2\)) in which we would like the connected components of the resulting graph to belong to, and design simpler and more efficient FPT algorithms. We design two general algorithms for pairs of graph classes (possibly having infinite forbidden sets) satisfying certain conditions on their forbidden sets. These algorithms cover a number of pairs of popular graph classes.
Our algorithms make non-trivial use of the branching technique and as black box, FPT algorithms for deletion to individual graph classes.
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- 1.
A hereditary graph class is a class of graphs that is closed under induced subgraphs.
- 2.
\({\mathcal O}^*\) notation suppresses polynomial factors.
- 3.
Proofs of Theorems and Lemmas marked \(\star \) are moved to the full version due to lack of space.
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Jacob, A., Majumdar, D., Raman, V. (2021). Faster FPT Algorithms for Deletion to Pairs of Graph Classes. In: Bampis, E., Pagourtzis, A. (eds) Fundamentals of Computation Theory. FCT 2021. Lecture Notes in Computer Science(), vol 12867. Springer, Cham. https://doi.org/10.1007/978-3-030-86593-1_22
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DOI: https://doi.org/10.1007/978-3-030-86593-1_22
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