面向块图与严格弦图的边修改问题的多项式核
Polynomial kernels for edge modification problems towards block and strictly chordal graphs
AI总结：
本文研究面向块图和严格弦图的边修改问题，证明大多数变体NP完全，并分别给出规模为O(k^2)、O(k^3)和O(k^4)的多项式顶点核。
摘要：
我们考虑面向块图和严格弦图的边修改问题，其中给定一个无向图 $G = (V,E)$ 和一个整数 $k \in \mathbb{N}$，要求通过编辑（添加或删除）$G$ 中至多 $k$ 条边来得到一个块图或一个严格弦图。这些问题的补全变体和删除变体定义类似，前者只允许添加边，后者只允许删除边。块图是一类被广泛研究的图，并具有多种刻画，例如它们是无钻石的弦图。严格弦图，也称为块复制图，是块图的自然推广，其中可以添加割点的真孪生顶点。严格弦图恰好是无 dart 和无 gem 的弦图。我们证明了这些问题的大多数变体是 NP 完全的，并为 Block Graph Editing 和 Block Graph Deletion 提供了 $O(k^2)$ 个顶点的核，为 Strictly Chordal Completion 和 Strictly Chordal Deletion 提供了 $O(k^3)$ 个顶点的核，并为 Strictly Chordal Editing 提供了一个 $O(k^4)$ 个顶点的核。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
