Abstract
Given a non-negative integer $j$ and a positive integer $k$, a $j$-restricted $k$-matching in a simple undirected graph is a $k$-matching, so that each of its connected components has at least $j+1$ edges. The maximum non-negative node weighted $j$-restricted $k$-matching problem was recently studied by Li who gave a polynomial-time algorithm and a min-max theorem for $0 \leqslant j < k$, and also proved the NP-hardness of the problem with unit node weights and $2 \leqslant k \leqslant j$. In this paper we derive an Edmonds–Gallai-type decomposition theorem for the $j$-restricted $k$-matching problem with $0 \leqslant j < k$, using the analogous decomposition for $k$-piece packings given by Janata, Loebl and Szabó, and give an alternative proof to the min-max theorem of Li.
Highlights
In this paper all graphs are simple and undirected
An F-packing M is perfect if V (M ) = V (G)
In all known polynomial F-packing problems with K2 ∈ F it holds that the electronic journal of combinatorics 27(1) (2020), #P1.15 each maximal F-packing is maximum too; those node sets which can be covered by an Fpacking form a matroid, and the analogue of the classical Edmonds–Gallai decomposition theorem for matchings holds
Summary
In this paper all graphs are simple and undirected. Given a set F of graphs, an Fpacking of a graph G is a subgraph M of G such that each connected component of M is isomorphic to a member of F. The most prominent example is the fact that the critical graphs with respect to the j-restricted k-matching problem are critical with respect to the k-piece packing problem (the role of critical graphs will be clear from the Edmonds–Gallai-type decomposition Theorems 3 and 10) This connection makes it possible to translate the analysis on k-piece packings to j-restricted k-matchings, the electronic journal of combinatorics 27(1) (2020), #P1.15 and to prove analogous results. Exploiting this relationship, in this paper we give an alternative proof to Theorem 6 of Li [14].
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