On the Monitoring-Edge-Geodetic Numbers of Line Graphs

IF 0.5 Q4 COMPUTER SCIENCE, THEORY & METHODS

JOURNAL OF INTERCONNECTION NETWORKS Pub Date : 2023-10-20 DOI:10.1142/s0219265923500251

Gemaji Bao, Chenxu Yang, Zhiqiang Ma, Zhen Ji, Xin Xu, Peiyao Qin

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引用次数: 0

Abstract

For a vertex set [Formula: see text], we say that [Formula: see text] is a monitoring-edge-geodetic set (MEG-set for short) of graph [Formula: see text], that is, some vertices of [Formula: see text] can monitor an edge of the graph, if and only if we can remove that edge would change the distance between some pair of vertices in the set. The monitoring-edge-geodetic number [Formula: see text] of a graph [Formula: see text] is defined as the minimum cardinality of a monitoring-edge-geodetic set of [Formula: see text]. The line graph [Formula: see text] of [Formula: see text] is the graph whose vertices are in one-to-one correspondence with the edges of [Formula: see text], that is, if two vertices are adjacent in [Formula: see text] if and only if the corresponding edges have a common vertex in [Formula: see text]. In this paper, we study the relation between [Formula: see text] and [Formula: see text], and prove that [Formula: see text]. Next, we have determined the exact values for a MEG-set of some special graphs and their line graphs. For a graph [Formula: see text] and its line graph [Formula: see text], we prove that [Formula: see text] can be arbitrarily large.

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关于线形图的监测边测地数

对于一个顶点集[公式:见文]，我们说[公式:见文]是图[公式:见文]的一个监控边测地集(简称meg集)，即[公式:见文]的一些顶点可以监控图的某条边，当且仅当我们可以移除这条边会改变集合中某些顶点对之间的距离。图[公式:见文]的监测边测地线数[公式:见文]定义为[公式:见文]的监测边测地线集合的最小基数。[公式:见文]的直线图[公式:见文]是顶点与[公式:见文]的边一一对应的图，即当且仅当[公式:见文]中两个顶点相邻时，当对应的边有[公式:见文]中的一个公共顶点。本文研究了[公式:见文]与[公式:见文]之间的关系，证明了[公式:见文]。接下来，我们确定了一些特殊图及其线形图的meg集的确切值。对于一个图[公式:见文]和它的线图[公式:见文]，我们证明了[公式:见文]可以任意大。

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.