Nordhaus–Gaddum-Type Results for the k-Independent Number of Graphs

IF 0.5 Q4 COMPUTER SCIENCE, THEORY & METHODS

JOURNAL OF INTERCONNECTION NETWORKS Pub Date : 2023-06-10 DOI:10.1142/s021926592350007x

Zhao Wang, Hongfang Liu, Yuhu Liu

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引用次数: 0

Abstract

The concept of [Formula: see text]-independent set, introduced by Fink and Jacobson in 1986, is a natural generalization of classical independence set. A k-independent set is a set of vertices whose induced subgraph has maximum degree at most [Formula: see text]. The k-independence number of [Formula: see text], denoted by [Formula: see text], is defined as the maximum cardinality of a [Formula: see text]-independent set of [Formula: see text]. As a natural counterpart of the [Formula: see text]-independence number, we introduced the concept of [Formula: see text]-edge-independence number. An edge set [Formula: see text] in [Formula: see text] is called k-edge-independent if the maximum degree of the subgraph induced by the edges in [Formula: see text] is less or equal to [Formula: see text]. The k-edge-independence number, denoted [Formula: see text], is defined as the maximum cardinality of a [Formula: see text]-edge-independent set. In this paper, we study the Nordhaus–Gaddum-type results for the parameter [Formula: see text] and [Formula: see text]. We obtain sharp upper and lower bounds of [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] for a graph [Formula: see text] of order [Formula: see text]. Some graph classes attaining these bounds are also given.

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图的k独立数的nordhaus - gaddum型结果

Fink和Jacobson于1986年提出的[公式:见文]独立集的概念是对经典独立集的自然推广。k无关集是指诱导子图最多具有最大度的顶点集合[公式:见文]。[公式:见文]的k独立数，用[公式:见文]表示，定义为[公式:见文]的[公式:见文]的[公式:见文]独立集合的最大基数。作为[公式:见文]-独立数的自然对应，我们引入了[公式:见文]-边缘独立数的概念。如果由[公式:见文]中的边引起的子图的最大程度小于或等于[公式:见文]，则称为[公式:见文]中的边集[公式:见文]中的边集[k-edge-independent]。k-边无关数，记为[公式:见文]，定义为[公式:见文]-边无关集的最大基数。本文研究了参数[公式:见文]和[公式:见文]的nordhaus - gaddum型结果。对于有序的[公式:见文]图[公式:见文]，我们得到了[公式:见文]，[公式:见文]，[公式:见文]，[公式:见文]，[公式:见文]，[公式:见文]的清晰的上界和下界。并给出了达到这些边界的一些图类。

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 INTERCONNECTION NETWORKS COMPUTER SCIENCE, THEORY & METHODS-

自引率

14.30%

发文量

121

期刊介绍： The Journal of Interconnection Networks (JOIN) is an international scientific journal dedicated to advancing the state-of-the-art of interconnection networks. The journal addresses all aspects of interconnection networks including their theory, analysis, design, implementation and application, and corresponding issues of communication, computing and function arising from (or applied to) a variety of multifaceted networks. Interconnection problems occur at different levels in the hardware and software design of communicating entities in integrated circuits, multiprocessors, multicomputers, and communication networks as diverse as telephone systems, cable network systems, computer networks, mobile communication networks, satellite network systems, the Internet and biological systems.