Many-to-Many Disjoint Paths in Augmented Cubes With Exponential Fault Edges

Mingzu Zhang,Wenhuan Ma,Tianlong Ma

doi:10.1109/access.2021.3095370

Mingzu Zhang, Wenhuan Ma + Show 1 more

Open Access

https://doi.org/10.1109/access.2021.3095370

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Abstract

It is common knowledge that edge disjoint paths have close relationship with the edge connectivity. Motivated by the well-known Menger theorem, we find that the maximum cardinality of edge disjoint paths connecting any two disjoint connected subgraphs with g vertices in G can also define by the minimum modified edge-cut, called the g-extra edge-connectivity of G (λ g (G)). It is the cardinality of the minimum set of edges in G, if such a set exists, whose deletion disconnects G and leaves every remaining component with at least g vertices. The n-dimensional augmented cube AQ n is a variant of hypercube Q n . In this paper, we observe that the g-extra edge-connectivity of the augmented cube for some exponentially large enough g exists a concentration behavior, for about 72.22 percent values of g ≤ 2 n-1 , and that the g-extra edge-connectivity of AQ n ( n ≥ 3) concentrates on ⌈n/2⌉-1 special values. Specifically, we prove that the exact value of g-extra edge-connectivity of augmented cube is a constant 2(⌈n/2⌉-r)2 ⌈n/2⌉+r for each integer 2 ⌈n/2⌉+r -l r ≤ g ≤ 2 ⌈n/2⌉+r , where n ≥ 3, r=1, 2, ..., ⌈ n/2⌉-1 and l r =\frac 2 2r+1 -23 if n is odd and l r =\frac 2 2r+2 -43 if n is even. The above upper and lower bounds of g are sharp. Moreover, we also obtain the exponential edge disjoint paths in AQ n with edge faults.

Highlights

T HE edge disjoint path problems are applicable in many areas such as software testing, database design and code optimization
Instead of considering the paired and unpaired many-to-many disjoint paths cover problem [9], [13], [20], [30], we focus on the problem of evaluation maximum number of many-to-many edge disjoint paths of a graph
We aim to find the maximum number of edge disjoint paths linking any two disjoint connected subgraphs with just g vertices in G

Summary

INTRODUCTION

T HE edge disjoint path problems are applicable in many areas such as software testing, database design and code optimization. Menger’s theorem is a description of the edge connectivity in finite graphs according to the maximum number of edge-disjoint paths that can be found between any two distinct pairs of vertices. Zhang et al.: Many-to-Many Disjoint Paths in Augmented Cubes with Exponential Fault Edges. Current authors connectivity κ(G) or edge connectivity λ(G), introduced by Menger [15], is defined as the minimum number of vertices or edges whose deletion disconnects the graph G. By the well-known Menger theorem, the maximum number of edge disjoint paths connecting any two disjoint connected subgraphs with g vertices in G can define by the minimum modified edge-cut, called (λg(G)). We study the g-extra edge-connectivity of the augmented cube AQn. we obtain the edge disjoint paths with edge faults.

PRELIMINARIES

1: Suppose 2 n 2

CONCLUDING REMARKS