TY - JOUR

T1 - Connectivity for some families of composition networks

AU - Chen, Hong

AU - Chen, Meirun

AU - Habib, Michel

AU - Lin, Cheng Kuan

N1 - Publisher Copyright:
© 2022

PY - 2022/6/24

Y1 - 2022/6/24

N2 - Connectivity of a connected graph G, κ(G), is an important index in exploring network topology which is the minimal number of vertices that need to be removed to separate G into disconnected or trivial. Let G0,G1,…,Gm−1 be m connected graphs of the same order. A matching composition network G is constructed by adding an arbitrary perfect matching between G0 and G1. For m≥3, a cycle composition network H is constructed by adding an arbitrary perfect matching between Gi and Gi+1(modm) for each 0≤i≤m−1. This construction has been so widely used in literature to build networks in which fault diagnosability can be studied, that it is worth to study their connectivity in detail, this is the main purpose of this paper. In this paper, we determine (1) κ(G)=δ(G) if κ(G0)+κ(G1)≥δ(G); otherwise, κ(G)≥κ(G0)+κ(G1), and (2) κ(H)=δ(H) if ∑i=0m−1κ(Gi)≥δ(H); otherwise, κ(H)≥∑i=0m−1κ(Gi). Examples show those bounds are tight. We then generalize these examples to a general composition using matchings on which we propose a conjecture on the connectivity and prove it for an important particular case.

AB - Connectivity of a connected graph G, κ(G), is an important index in exploring network topology which is the minimal number of vertices that need to be removed to separate G into disconnected or trivial. Let G0,G1,…,Gm−1 be m connected graphs of the same order. A matching composition network G is constructed by adding an arbitrary perfect matching between G0 and G1. For m≥3, a cycle composition network H is constructed by adding an arbitrary perfect matching between Gi and Gi+1(modm) for each 0≤i≤m−1. This construction has been so widely used in literature to build networks in which fault diagnosability can be studied, that it is worth to study their connectivity in detail, this is the main purpose of this paper. In this paper, we determine (1) κ(G)=δ(G) if κ(G0)+κ(G1)≥δ(G); otherwise, κ(G)≥κ(G0)+κ(G1), and (2) κ(H)=δ(H) if ∑i=0m−1κ(Gi)≥δ(H); otherwise, κ(H)≥∑i=0m−1κ(Gi). Examples show those bounds are tight. We then generalize these examples to a general composition using matchings on which we propose a conjecture on the connectivity and prove it for an important particular case.

KW - Connectivity

KW - Cycle composition networks

KW - Matching composition networks

UR - http://www.scopus.com/inward/record.url?scp=85130364297&partnerID=8YFLogxK

U2 - 10.1016/j.tcs.2022.04.036

DO - 10.1016/j.tcs.2022.04.036

M3 - Article

AN - SCOPUS:85130364297

VL - 922

SP - 361

EP - 367

JO - Theoretical Computer Science

JF - Theoretical Computer Science

SN - 0304-3975

ER -