Petersen Graph is Uniformly Most-Reliable

Rela, Guillermo; Robledo, Franco; Romero, Pablo

doi:10.1007/978-3-319-72926-8_35

Cited by 8 publications

(7 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…More recently, Romero (2017) formally proved using iterative augmentation that M 4 , known as Wagner graph, is uniformly most-reliable as well. In this sense, Möbius graphs apparently generalize the particular result for K 3,3 and K 4 , however M 5 is not UMRG, since Petersen graph is UMRG; see Rela et al (2018).…”

Section: Finding Sparse Umrgmentioning

confidence: 63%

“…Since Heawood and Mobius-Kantor meet the necessary criterion from Proposition 1 and also have largest girth, it is conjectured that these graphs also belong to the set of UMRGs. Rela et al (2018) proved that Petersen graph is UMRG. Furthermore, the only non-trivial 4-regular UMRG known so far is the complement of a cycle with seven nodes H = C 7 also known in the literature as an odd-antihole (Rela et al, 2019).…”

Section: Recent Progressmentioning

confidence: 99%

See 1 more Smart Citation

Finding uniformly most reliable graphs by counting trivial cuts

Canale¹,

Rela²,

Robledo³

et al. 2020

Preprint

Self Cite

View full text Add to dashboard Cite

In network design, the all-terminal reliability maximization is of paramount importance. In this classical setting, we assume a simple graph with perfect nodes but independent edge failures with identical probability ρ. The goal is to communicate n terminals using e edges, in such a way that the connectedness probability of the resulting random graph is maximum. A graph with n nodes and e edges that meets the maximum reliability property for all ρ ∈ (0, 1) is called uniformly most-reliable (n, e)-graph (UMRG). The discovery of these graphs is a challenging problem that involves an interplay between extremal graph theory and computational optimization. Recent works confirm the existence of special cubic UMRGs, such as Wagner, Petersen and Yutsis graphs, and a 4-regular graph H = C7. In a foundational work in the field, Boesch. et. al. state with no proof that the bipartite complete graph K4,4 is UMRG. In this paper, we revisit the breakthroughs in the theory of UMRG. A simple methodology to determine UMRGs based on counting trivial cuts is presented. Finally, we test this methodology to mathematically prove that the complete bipartite graph K4,4 is UMRG.

show abstract

Section: Finding Sparse Umrgmentioning

confidence: 63%

Section: Recent Progressmentioning

confidence: 99%

Finding uniformly most reliable graphs by counting trivial cuts

Canale¹,

Rela²,

Robledo³

et al. 2020

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Wagner graph [11] (10, 15) Petersen graph [13] Table 1: Uniformly most reliable graphs known in G(n, m), where n denotes the number of vertices and m the number of edges.…”

Section: Uniformly Most Reliable Graphsmentioning

confidence: 99%

“…The fair cake-cutting graphs F CG n,c are similar to the family of graphs that are a solution of the following augmentation problem: Starting from the cycle graph C n , add a single edge at each step, in order to maximize the reliability of the resulting graph. Romero (see [11,13]) finds the sequence of graphs {G (i) } i=0,..., n 2 with G (0) = C n such that G (i+1) = G (i) ∪ {e i+1 } gives the best augmentation. This process (called also the fair cake-cutting process) ends with the circulant graph with steps 1 and n 2 , that is, a cubic hamiltonian graph where every vertex is joined to its opposite vertex in the hamiltonian cycle.…”

Section: The Fair Cake-cutting Graph F Cg Ncmentioning

confidence: 99%

See 1 more Smart Citation

Network reliability in hamiltonian graphs

Llagostera¹,

López²,

Comas³

2020

Preprint

View full text Add to dashboard Cite

The reliability polynomial of a graph gives the probability that a graph remains operational when all its edges could fail independently with a certain fixed probability. In general, the problem of finding uniformly most reliable graphs inside a family of graphs, that is, one graph whose reliability is at least as large as any other graph inside the family, is very difficult. In this paper, we study this problem in the family of graphs containing a hamiltonian cycle.

show abstract

Uniformly optimally reliable graphs: A survey

Romero

2021

Networks

Self Cite

View full text Add to dashboard Cite

Which is the most reliable graph with n nodes and m edges? This celebrated problem has several aspects, according to the notion of optimality (in a local or uniform sense), failure type (either nodes or edges), or reliability model (all‐terminal connectedness, two‐terminal or multiterminal setting). This article presents a chronological survey of the multiple proposals to address the problem, together with recent trends and enigmatic conjectures posed decades ago that promote further research.

show abstract

Petersen Graph is Uniformly Most-Reliable

Cited by 8 publications

References 16 publications

Finding uniformly most reliable graphs by counting trivial cuts

Finding uniformly most reliable graphs by counting trivial cuts

Network reliability in hamiltonian graphs

Uniformly optimally reliable graphs: A survey

Contact Info

Product

Resources

About