Connectedness of finite distance graphs

Gómez-Pérez, Domingo; Gutiérrez, Jaime; Ibeas, Ángel

doi:10.1002/net.21465

Cited by 2 publications

(1 citation statement)

References 13 publications

(16 reference statements)

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“…Naturally, when a network exhibits special structure, one can hope for faster algorithms, more accurate estimates, and different bounds. These themes are well‐represented in Networks , by studies on trees [10], planar graphs [28], partial k ‐trees [150], complete graphs [141], interval and permutation graphs [1], distance graphs [104], and/or graphs [149], and circulant graphs [148].…”

Section: Reliability: the Roads Traveledmentioning

confidence: 99%

Network reliability: Heading out on the highway

et al. 2020

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A variety of probabilistic notions of network reliability of graphs and digraphs have been proposed and studied since the early 1950s. Although grounded in the engineering and logistics of network design and analysis, the research also spans pure and applied mathematics, with connections to areas as diverse as combinatorics and graph theory, combinatorial enumeration, optimization, probability theory, real and complex analysis, algebraic topology, commutative algebra, the design and analysis of algorithms, and computational complexity. In this paper we describe the landscape of various notions of network reliability, the roads well traveled, and some that appear likely to lead to meaningful and important journeys.

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Section: Reliability: the Roads Traveledmentioning

confidence: 99%

Network reliability: Heading out on the highway

et al. 2020

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On Hamiltonian Paths and Cycles in Sufficiently Large Distance Graphs

Löwenstein¹,

Rautenbach²,

Soták³

2014

Discrete Mathematics &Amp; Theoretical Computer Science

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Graph Theory International audience For a positive integer n∈ℕ and a set D⊆ ℕ, the distance graph GnD has vertex set { 0,1,\textellipsis,n-1} and two vertices i and j of GnD are adjacent exactly if |j-i|∈D. The condition gcd(D)=1 is necessary for a distance graph GnD being connected. Let D={d1,d2}⊆ℕ be such that d1>d2 and gcd(d1,d2)=1. We prove the following results. If n is sufficiently large in terms of D, then GnD has a Hamiltonian path with endvertices 0 and n-1. If d1d2 is odd, n is even and sufficiently large in terms of D, then GnD has a Hamiltonian cycle. If d1d2 is even and n is sufficiently large in terms of D, then GnD has a Hamiltonian cycle.

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Connectedness of finite distance graphs

Abstract: We describe a polynomial‐time algorithm for deciding whether a given distance graph with a finite number of vertices is connected. This problem was conjectured to be NP‐hard in Draque Penso et al. © 2012 Wiley Periodicals, Inc. NETWORKS, 2012

Cited by 2 publications

References 13 publications

Network reliability: Heading out on the highway

Network reliability: Heading out on the highway

On Hamiltonian Paths and Cycles in Sufficiently Large Distance Graphs

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