Reliability Analysis of Alternating Group Graphs and Split-Stars

Author:

Gu Mei-Mei12,Hao Rong-Xia1,Chang Jou-Ming3

Affiliation:

1. Department of Mathematics, Beijing Jiaotong University, Beijing 100044, China

2. Faculty of Mathematics and Physics, Charles University, Prague 11800, Czech Republic

3. Institute of Information and Decision Sciences, National Taipei University of Business, Taipei, 10051, Taiwan

Abstract

Abstract Given a connected graph $G$ and a positive integer $\ell $, the $\ell $-extra (resp. $\ell $-component) edge connectivity of $G$, denoted by $\lambda ^{(\ell )}(G)$ (resp. $\lambda _{\ell }(G)$), is the minimum number of edges whose removal from $G$ results in a disconnected graph so that every component has more than $\ell $ vertices (resp. so that it contains at least $\ell $ components). This naturally generalizes the classical edge connectivity of graphs defined in term of the minimum edge cut. In this paper, we proposed a general approach to derive component (resp. extra) edge connectivity for a connected graph $G$. For a connected graph $G$, let $S$ be a vertex subset of $G$ for $G\in \{\Gamma _{n}(\Delta ),AG_n,S_n^2\}$ such that $|S|=s\leq |V(G)|/2$, $G[S]$ is connected and $|E(S,G-S)|=\min \limits _{U\subseteq V(G)}\{|E(U, G-U)|: |U|=s, G[U]\ \textrm{is connected}\ \}$, then we prove that $\lambda ^{(s-1)}(G)=|E(S,G-S)|$ and $\lambda _{s+1}(G)=|E(S,G-S)|+|E(G[S])|$ for $s=3,4,5$. By exploring the reliability analysis of $AG_n$ and $S_n^2$ based on extra (component) edge faults, we obtain the following results: (i) $\lambda _3(AG_n)-1=\lambda ^{(1)}(AG_n)=4n-10$, $\lambda _4(AG_n)-3=\lambda ^{(2)}(AG_n)=6n-18$ and $\lambda _5(AG_n)-4=\lambda ^{(3)}(AG_n)=8n-24$; (ii) $\lambda _3(S_n^2)-1=\lambda ^{(1)}(S_n^2)=4n-8$, $\lambda _4(S_n^2)-3=\lambda ^{(2)}(S_n^2)=6n-15$ and $\lambda _5(S_n^2)-4=\lambda ^{(3)}(S_n^2)=8n-20$. This general approach maybe applied to many diverse networks.

Funder

Czech Operational Programme Research, Development and Education project, International Mobility of Researchers at Charles University

China Postdoctoral Science Foundation

National Natural Science Foundation of China

111 Project of China

Ministry of Science and Technology

Publisher

Oxford University Press (OUP)

Subject

General Computer Science

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