An algorithm for computing simple k-factors

Meijer, Henk; Núòez-Rodríguez, Yurai; Rappaport, David

doi:10.1016/j.ipl.2009.02.013

Cited by 7 publications

(8 citation statements)

References 12 publications

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“…In this context, our notion of k-matchings is equivalent to so-called component factors while perfect k-matchings coincide with the well-known notion of k-factors which are k-regular spanning graphs. While it is an easy task to verify the existence and to find a perfect k-matching (if one exists) [3,7,24], it is NP-complete to determine whether there is a nonempty k-matchings in a graph for all k ≥ 3 [7,26,32].…”

Section: Preliminariesmentioning

confidence: 99%

“…A k-matching M is perfect, if each vertex is incident to some edge of M , and thus, F is a k-regular spanning graph of G. Determining whether a perfect k-matching exists and, in the affirmative case, computing a perfect k-matching can be done in polynomial-time [3,7,24]. In contrast, the problem of determining whether a given graph has a non-empty k-matching, is NP-complete for all integer k ≥ 3 [7,26,32].…”

Section: Introductionmentioning

confidence: 99%

“…In contrast, the problem of determining whether a given graph has a non-empty k-matching, is NP-complete for all integer k ≥ 3 [7,26,32]. Perfect k-matchings are also known as k-factors [19,20,24,27,29,[33][34][35]39]; a term that we do not use in order to avoid confusion with the factors G and H of a product G H. The k-matching number m k (G) is the number of edges contained in a maximum-sized k-matching of G.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Lindeberg¹,

Hellmuth²

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…A k-matching M is perfect, if each vertex is incident to some edge of M , and thus, F is a k-regular spanning graph of G. Determining whether a perfect k-matching exists and, in the affirmative case, computing a perfect k-matching can be done in polynomial-time [2,6,24]. In contrast, the problem of determining whether a given graph has a non-empty k-matching, is NP-complete for all integers k ≥ 3 [6,26,33].…”

Section: Introductionmentioning

confidence: 99%

Construction of k-matchings in graph products

Lindeberg

Hellmuth

2022

Art Discrete Appl. Math.

View full text Add to dashboard Cite

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 in 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 different polynomial-time constructions of k-matchings in a graph product G ⋆ H that are based on k G -matchings 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 weak-homomorphism 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 weakhomomorphism preserving k-matchings, however, can be constructed in our manner. We will, therefore, determine the size of maximum-sized elements among all weak-homomorphism preserving k-matching within the respective graph products, provided that the matchings in the factors satisfy some general assumptions.

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“…An r-factor of a graph is an r-regular spanning subgraph. A 2-factor is also called a cycle cover and has many applications in areas such as computer graphics and computational geometry [21], for instance for fast rendering of 3D scenes. In an edge-weighted graph, the problem of finding a cycle cover with minimum cost can be solved in polynomial-time [25].…”

Section: Introductionmentioning

confidence: 99%

Minimum Reload Cost Graph Factors

et al. 2021

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The concept of Reload cost in a graph refers to the cost that occurs while traversing a vertex via two of its incident edges. This cost is uniquely determined by the colors of the two edges. This concept has various applications in transportation networks, communication networks, and energy distribution networks. Various problems using this model are defined and studied in the literature. The problem of finding a spanning tree whose diameter with respect to the reload costs is smallest possible, the problems of finding a path, trail or walk with minimum total reload cost between two given vertices, problems about finding a proper edge coloring of a graph such that the total reload cost is minimized, the problem of finding a spanning tree such that the sum of the reload costs of all paths between all pairs of vertices is minimized, and the problem of finding a set of cycles of minimum reload cost, that cover all the vertices of a graph, are examples of such problems. In this work we focus on the last problem. Noting that a cycle cover of a graph is a 2-factor of it, we generalize the problem the the problem of finding an r-factor of minimum reload cost of an edge colored graph. We prove several NP-hardness results for special cases of the problem. Namely, bounded degree graphs, planar graphs, bounded total cost, and bounded number of distinct costs. For the special case of r = 2, our results imply an improved NP-hardness result. On the positive side, we present a polynomial-time solvable special case which provides a tight boundary between the polynomial and hard cases in terms of r and the and the maximum degree of the graph. We then investigate the parameterized complexity of the problem, prove W[1]-hardness results and present an FPT-algorithm.

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An algorithm for computing simple k-factors

Cited by 7 publications

References 12 publications

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

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

Construction of k-matchings in graph products

Minimum Reload Cost Graph Factors

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