A Validation of the Phenomenon of Linearly Many Faults on Burnt Pancake Graphs with Its Applications

Abstract

“Linearly many faults” is a phenomenon observed by Cheng and Lipták in which a specific structure emerges when a graph is disconnected and often occurs in various interconnection networks. This phenomenon means that if a certain number of vertices or edges are deleted from a graph, the remaining part either stays connected or breaks into one large component along with smaller components with just a few vertices. This phenomenon can be observed in many types of graphs and has important implications for network analysis and optimization. In this paper, we first validate the phenomenon of linearly many faults for surviving graph of a burnt pancake graph BPn when removing any edge subset with a size of approximately six times λ(BPn). For graph G, the ℓ-component edge connectivity denoted as λℓ(G) (resp., the ℓ-extra edge connectivity denoted as λ(ℓ)(G)) is the cardinality of a minimum edge subset S such that G−S is disconnected and has at least ℓ components (resp., each component of G−S has at least ℓ+1 vertices). Both λℓ(G) and eλ(ℓ)(G) are essential metrics for network reliability assessment. Specifically, from the property of “linearly many faults”, we may further prove that λ5(BPn)=λ(3)(BPn)+3=4n−3 for n⩾5; λ6(BPn)=λ(4)(BPn)+4=5n−4 and λ7(BPn)=λ(5)(BPn)+5=6n−5 for n⩾6.