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 Edition 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 Edition.
翻译:我们考虑对区块图和严格的chordal 图形的边缘修改问题,即给一个未定向的图形$G = (V,E) 美元和整金 $k = commathb{N}$ $,并试图从$G$中编辑(增加或删除) 最多为$G$的边缘,以获得区块图或严格的chodal 图。这些问题的完成和删除变体定义相似,仅允许前一图和后一图的边缘添加。块图是一个研究周密的图表类别,并接受若干特性,例如,它们是无钻石的chordal 图形。严格的chordal 图表,也被称为区块重复图,是块图的自然概括,可以添加真正的切口双胞。严格意义上的Choordal 图形是完全的 dart 和 宝石无色标图。我们证明这些问题的大多数变体的NP(NP) 的完整性,并且为 Streal- $O (k2) 美元 和 Streal- Challifrial- Challigal- Chalim andal- 和 Drealvieward Strealdal- Chaldaldalphalphal和 。