Polynomial kernels for edge modification problems towards block and strictly chordal graphs

TL;DR

This paper studies edge modification problems to transform graphs into block or strictly chordal graphs, providing complexity results and polynomial kernelizations for various problem variants.

Contribution

It establishes NP-completeness for most variants and offers new polynomial kernelizations for block and strictly chordal graph editing problems.

Findings

NP-completeness for most variants

Polynomial kernels for block and strictly chordal editing problems

Efficient preprocessing algorithms for graph modification tasks

Abstract

We consider edge modification problems towards block and strictly chordal graphs, where one is given an undirected graph $G = (V,E)$ and an integer $k \in \mathbb{N}$ and seeks to edit (add or delete) at most $k$ edges from $G$ to obtain a block graph or a strictly chordal graph. The completion and deletion variants of these problems are defined similarly by only allowing edge additions for the former and only edge deletions for the latter. Block graphs are a well-studied class of graphs and admit several characterizations, e.g. they are diamond-free chordal graphs. Strictly chordal graphs, also referred to as block duplicate graphs, are a natural generalization of block graphs where one can add true twins of cut-vertices. Strictly chordal graphs are exactly dart and gem-free chordal graphs. We prove the NP-completeness for most variants of these problems and provide $O(k^2)$ vertex-kernels for Block Graph Editing and Block Graph Deletion, $O(k^3)$ vertex-kernels for Strictly Chordal Completion and Strictly Chordal Deletion and a $O(k^4)$ vertex-kernel for Strictly Chordal Editing.