The p-restricted edge-connectivity of Kneser graphs

Balbuena, C.; Marcote, X.

doi:10.1016/j.amc.2018.09.072

Cited by 5 publications

(2 citation statements)

References 27 publications

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“…In this case, κ is strictly greater than the connectivity κ = κ(G) of G; otherwise κ = κ. Some examples of graph classes which have been analysed for their super-connectivity are circulant graphs [3], products of various graphs (see [7,8,12,19], and the references therein), hypercubes [13,24,25], generalized Petersen graphs [4], Johnson graphs [5] and Kneser graphs [1,6].…”

Section: Introductionmentioning

confidence: 99%

The super-connectivity of odd graphs and of their kronecker double cover

Ek̇inċi

Gauci

2021

RAIRO-Oper. Res.

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The study of connectivity parameters forms an integral part of the research conducted in establishing the fault tolerance of networks. A number of variations on the classical notion of connectivity have been proposed and studied. In particular, the super--connectivity asks for the minimum number of vertices that need to be deleted from a graph in order to disconnect the graph without creating isolated vertices. In this work, we determine this value for two closely related families of graphs which are considered as good models for networks, namely the odd graphs and their Kronecker double cover. The odd graphs are constructed by taking all possible subsets of size $k$ from the set of integers $\{1,\ldots,2k+1\}$ as vertices, and defining two vertices to be adjacent if the corresponding $k$-subsets are disjoint; these correspond to the Kneser graphs $KG(2k+1,k)$. The Kronecker double cover of a graph $G$ is formed by taking the Kronecker product of $G$ with the complete graph on two vertices; in the case when $G$ is $KG(2k+1,k)$, the Kronecker double cover is the bipartite Kneser graph $H(2k+1,k)$. We show that in both instances, the super--connectivity is equal to $2k$.

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Section: Introductionmentioning

confidence: 99%

The super-connectivity of odd graphs and of their kronecker double cover

Ek̇inċi

Gauci

2021

RAIRO-Oper. Res.

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“…A graph G is super-connected if every minimum vertex-cut is composed of the neighbourhood N G (x) of a vertex x ∈ V (G), where N G (x) = {y ∈ V (G) : xy ∈ E(G)}. Some examples of graph classes which have been analysed for their super-connectivity are circulant graphs [4], hypercubes [17,29,30], products of various graphs (see [7,9,16,22], and the references therein), generalized Petersen graphs [5], minimal Cayley graphs [18] and Kneser graphs [3,6]. In this work we analyse and establish the super-connectivity of Johnson graphs.…”

Section: Introductionmentioning

confidence: 99%

The super-connectivity of Johnson graphs

Ekinci,

Gauci

2019

Preprint

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For positive integers n, k and t, the uniform subset graph G(n, k, t) has all k-subsets of {1, 2, . . . , n} as vertices and two k-subsets are joined by an edge if they intersect at exactly t elements. The Johnson graph J(n, k) corresponds to G(n, k, k − 1), that is, two vertices of J(n, k) are adjacent if the intersection of the corresponding k-subsets has size k − 1. A super vertex-cut of a connected graph is a set of vertices whose removal disconnects the graph without isolating a vertex and the superconnectivity is the size of a minimum super vertex-cut. In this work, we fully determine the super-connectivity of the family of Johnson graphs J(n, k) for n ≥ k ≥ 1.

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