Building large k-cores from sparse graphs

IF 1.1 3区计算机科学 Q1 BUSINESS, FINANCE Journal of Computer and System Sciences Pub Date : 2023-03-01 DOI:10.1016/j.jcss.2022.10.002

Fedor V. Fomin , Danil Sagunov , Kirill Simonov

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Abstract

A k-core of a graph G is the maximal induced subgraph in which every vertex has degree at least k. In the Edge k-Core optimization problem, we are given a graph G and integers k, b and p. The task is to ensure that the k-core of G has at least p vertices, by adding at most b edges. While Edge k-Core is known to be computationally hard in general, we show that there are efficient algorithms when the k-core has to be constructed from a sparse graph with some structural properties. Our results are as follows.

•
When the input graph is a forest, Edge k-Core is solvable in polynomial time.
•
Edge k-Core is fixed-parameter tractable (FPT) when parameterized by the minimum size of a vertex cover in the input graph.
•
Edge k-Core is $FPT$ when parameterized by the treewidth of the graph plus k.

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从稀疏图构建大型k核

图G的k-核是每个顶点的度至少为k的最大诱导子图。在边k-核优化问题中，我们给出了一个图G和整数k、b和p。任务是通过添加最多b条边来确保G的k-核心至少有p个顶点。虽然边缘k-Core通常在计算上是困难的，但我们证明了当k-Core必须由具有某些结构属性的稀疏图构造时，存在有效的算法。我们的结果如下。•当输入图是森林时，边k-Core在多项式时间内是可解的。•当通过输入图中顶点覆盖的最小大小进行参数化时，边k-Core是固定参数可处理的（FPT）。•当通过图的树宽度加上k来参数化时，边k-Core是FPT。

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 Computer and System Sciences 工程技术-计算机：理论方法

CiteScore

3.70

自引率

0.00%

发文量

审稿时长

68 days

期刊介绍： The Journal of Computer and System Sciences publishes original research papers in computer science and related subjects in system science, with attention to the relevant mathematical theory. Applications-oriented papers may also be accepted and they are expected to contain deep analytic evaluation of the proposed solutions. Research areas include traditional subjects such as: • Theory of algorithms and computability • Formal languages • Automata theory Contemporary subjects such as: • Complexity theory • Algorithmic Complexity • Parallel & distributed computing • Computer networks • Neural networks • Computational learning theory • Database theory & practice • Computer modeling of complex systems • Security and Privacy.