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

2022-01-31

arXiv.org

AI Summary

This paper studies the graph editing problem: given an undirected graph (G) and an integer (k), determine whether (G) can be transformed into a block graph or a strictly chordal graph using at most (k) edge additions, deletions, or both (with variants restricting to completion or deletion only). The authors develop the first polynomial-size vertex kernels for all five variants—block graph editing and deletion, and strictly chordal graph completion, deletion, and editing—achieving kernel sizes of (O(k^2)), (O(k^3)), and (O(k^4)), respectively. Their approach leverages structural characterizations—including forbidden induced subgraphs (e.g., diamonds, darts, gems), cut-vertices, and true twin relationships—combined with modular decomposition and safe contraction techniques. These kernels resolve long-standing open questions, as polynomial kernels were previously unknown for these problems. The results establish fixed-parameter tractability for all variants and provide tight preprocessing foundations for subsequent algorithm design.

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.

Problem

• Edge modification for block and strictly chordal graphs
• NP-completeness of most problem variants
• Polynomial vertex-kernels for editing and deletion problems

Innovation

• Develops polynomial kernels for graph editing.
• Focuses on block and strictly chordal graphs.
• Provides O(k^4) vertex-kernel for chordal editing.