The Spanning Connectivity of the Burnt Pancake Graphs

Chin, Cherng; Weng, Tien‐Hsiung; Hsu, Lih‐Hsing; Chiou, Shang-Chia

doi:10.1587/transinf.e92.d.389

Cited by 28 publications

(4 citation statements)

References 22 publications

(27 reference statements)

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“…Indeed, an external edge is an n-dimensional edge. Also, E j,k (BP n ) denotes the set of edges between BP j n and BP j n for j, k ∈ ⟨n⟩ with j ̸ = k. Lemma 2 (see [39,42,46]). For BP n , the following properties hold:…”

Section: Burnt Pancake Graphs Bp Nmentioning

confidence: 99%

“…For burnt pancake graphs, the earliest research mainly pursued their diameters, while the current research focuses on exploring fault tolerance [39,40] and diagnosis [40,41]. In addition, many diverse connectivities have been investigated in the literature, including spanning connectivity [42], structure connectivity [43], neighbor connectivity [44,45], and component connectivity [25,28]. Following the direction of probing connectivity, this paper first proves that when removing any edge subset with a size of approximately six times λ(BP n ), the surviving graph possesses the "linearly many faults" property.…”

Section: Introductionmentioning

confidence: 99%

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

Gu,

Yan,

Chang

2024

Mathematics

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“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.

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Section: Burnt Pancake Graphs Bp Nmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Gu,

Yan,

Chang

2024

Mathematics

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“…Table 1 shows the maximum distances with the corresponding patterns of pairs of faulty nodes. Table 1 Maximum distance from (1,2,3) to non-faulty nodes in B 3 with a pair of faulty nodes.…”

Section: Appendix Amentioning

confidence: 99%

“…In this study, we focus on a burnt pancake graph [2,3,5,7] with multiple faulty nodes, and propose a fault-tolerant routing algorithm in it. The burnt pancake graph is a variant of a pancake graph [1] that is derived from Cayley graphs.…”

Section: Introductionmentioning

confidence: 99%