Cutting a tree with subgraph complementation is hard, except for some small trees

IF 0.9 3区 数学 Q2 MATHEMATICS
Dhanyamol Antony, Sagartanu Pal, R. B. Sandeep, R. Subashini
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引用次数: 0

Abstract

For a graph property Π ${\rm{\Pi }}$ , Subgraph Complementation to Π ${\rm{\Pi }}$ is the problem to find whether there is a subset S $S$ of vertices of the input graph G $G$ such that modifying G $G$ by complementing the subgraph induced by S $S$ results in a graph satisfying the property Π ${\rm{\Pi }}$ . We prove that the problem of Subgraph Complementation to T $T$ -free graphs is NP-Complete, for T $T$ being a tree, except for 41 trees of at most 13 vertices (a graph is T $T$ -free if it does not contain any induced copies of T $T$ ). This result, along with the four known polynomial-time solvable cases (when T $T$ is a path on at most four vertices), leaves behind 37 open cases. Further, we prove that these hard problems do not admit any subexponential-time algorithms, assuming the Exponential-Time Hypothesis. As an additional result, we obtain that Subgraph Complementation to paw-free graphs can be solved in polynomial-time.

用子图互补法切割一棵树是很难的,除非是一些小树
对于一个图的属性而言,"补全子图"(Subgraph Complementation to)是一个问题,即找出输入图中是否存在这样一个顶点子集,即通过补全由其诱导的子图来进行修改,从而得到一个满足该属性的图。我们证明,对于树状图而言,子图补全到-free 图的问题是 NP-Complete(NP-Complete)的,但顶点数最多为 13 的 41 棵树(如果一个图不包含任何"-free "的诱导副本,则该图为-free)除外。这一结果,加上已知的四种多项式时间可解情况(当最多四个顶点上有一条路径时),留下了 37 种未解情况。此外,我们还证明,假设存在指数时间假说,这些难题不存在任何亚指数时间算法。作为附加结果,我们还得到了无爪图的子图补全可以在多项式时间内求解。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Journal of Graph Theory
Journal of Graph Theory 数学-数学
CiteScore
1.60
自引率
22.20%
发文量
130
审稿时长
6-12 weeks
期刊介绍: The Journal of Graph Theory is devoted to a variety of topics in graph theory, such as structural results about graphs, graph algorithms with theoretical emphasis, and discrete optimization on graphs. The scope of the journal also includes related areas in combinatorics and the interaction of graph theory with other mathematical sciences. A subscription to the Journal of Graph Theory includes a subscription to the Journal of Combinatorial Designs .
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