Construction of k-matchings and k-regular subgraphs in graph products
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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_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 weak-homomorphism preserving k-matchings, 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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