Matching preclusion for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="mml31" display="inline" overflow="scroll" altimg="si25.gif"><mml:mi>n</mml:mi></mml:math>-grid graphs

Ding, Qi; Zhang, Heping; Zhou, Hui

doi:10.1016/j.dam.2018.02.012

Cited by 4 publications

(2 citation statements)

References 17 publications

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“…1-matchings for graph products have been extensively studied, see e.g. [1,2,5,8,9,15,20,21,23,25,37,38]. Most of these results are concerned with quite restricted subclasses of the Cartesian product; restrictions that are needed to answer often quite difficult questions as e.g.…”

Section: Introductionmentioning

confidence: 99%

Construction of $k$-matchings and $k$-regular subgraphs in graph products

Lindeberg¹,

Hellmuth²

2021

Preprint

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A k-matching M of a graph G = (V, E) is a subset M ⊆ E such that each connected component in the subgraph F = (V, M ) of G is either a single-vertex graph or k-regular, i.e., each vertex has degree k. In this contribution, we are interested in k-matchings within the four standard graph products: the Cartesian, strong, direct and lexicographic product.As we shall see, the problem of finding non-empty k-matchings (k ≥ 3) in graph products is NP-complete. Due to the general intractability of this problem, we focus on distinct polynomial-time constructions of k-matchings in a graph product G H that are based on k Gmatchings M G and k H -matchings M H of its factors G and H, respectively. In particular, we are interested in properties of the factors that have to be satisfied such that these constructions yield a maximum k-matching in the respective products. Such constructions are also called "well-behaved" and we provide several characterizations for this type of k-matchings.Our specific constructions of k-matchings in graph products satisfy the property of being weak-homomorphism preserving, i.e., constructed matched edges in the product are never "projected" to unmatched edges in the factors. This leads to the concept of weakhomomorphism preserving k-matchings. Although the specific k-matchings constructed here are not always maximum k-matchings of the products, they have always maximum size among all weak-homomorphism preserving k-matchings. Not all weak-homomorphism preserving kmatchings, however, can be constructed in our manner. We will, therefore, determine the size of maximum-sized elements among all weak-homomorphims preserving k-matching within the respective graph products, provided that the matchings in the factors satisfy some general assumptions.

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Section: Introductionmentioning

confidence: 99%

Construction of $k$-matchings and $k$-regular subgraphs in graph products

Lindeberg¹,

Hellmuth²

2021

Preprint

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“…Until now, the matching preclusion number of numerous networks were calculated and the corresponding optimal solutions were obtained, such as the complete graph, the complete bipartite graph and the hypercube [6], Cayley graphs generated by 2-trees and hyper Petersen networks [10], Cayley graphs generalized by transpositions and (n, k)-star graphs [11], restricted HL-graphs and recursive circulant G(2 m , 4) [31], tori and related Cartesian products [12], (n, k)-bubble-sort graphs [13], balanced hypercubes [27], burnt pancake graphs [22], k-ary n-cubes [35], cube-connected cycles [25], vertex-transitive graphs [24], n-dimensional torus [23], binary de Bruijn graphs [26] and n-grid graphs [17]. For the conditional matching preclusion problem, it is solved for the complete graph, the complete bipartite graph and the hypercube [6], arrangement graphs [14], alternating group graphs and split-stars [15], Cayley graphs generated by 2-trees and the hyper Petersen networks [10], Cayley graphs generalized by transpositions and (n, k)-star graphs [11], burnt pancake graphs [8,22], balanced hypercubes [27], restricted HL-graphs and recursive circulant G(2 m , 4) [31], k-ary n-cubes [35], hypercube-like graphs [32] and cube-connected cycles [25].…”

Section: Introductionmentioning

confidence: 99%

Super edge-connectivity and matching preclusion of data center networks

Lü,

2018

Preprint

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Edge-connectivity is a classic measure for reliability of a network in the presence of edge failures. k-restricted edge-connectivity is one of the refined indicators for fault tolerance of large networks. Matching preclusion and conditional matching preclusion are two important measures for the robustness of networks in edge fault scenario. In this paper, we show that the DCell network D k,n is super-λ for k ≥ 2 and n ≥ 2, super-λ 2 for k ≥ 3 and n ≥ 2, or k = 2 and n = 2, and super-λ 3 for k ≥ 4 and n ≥ 3. Moreover, as an application of k-restricted edge-connectivity, we study the matching preclusion number and conditional matching preclusion number, and characterize the corresponding optimal solutions of D k,n . In particular, we have shown that D 1,n is isomorphic to the (n, k)-star graph S n+1,2 for n ≥ 2.

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On anti-Kekulé and s-restricted matching preclusion problems

Lü

Zhang

2023

J Comb Optim

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Matching preclusion for n-grid graphs

Cited by 4 publications

References 17 publications

Construction of $k$-matchings and $k$-regular subgraphs in graph products

Construction of $k$-matchings and $k$-regular subgraphs in graph products

Super edge-connectivity and matching preclusion of data center networks

On anti-Kekulé and s-restricted matching preclusion problems

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