An Edmonds-Gallai-Type Decomposition for the j-Restricted k-Matching Problem

The Edmonds-Gallai Decomposition for the k-Piece Packing Problem

Generalizing Kaneko’s long path packing problem, Hartvigsen, Hell and Szabó consider a new type of undirected graph packing problem, called the k-piece packing problem. A k-piece is a simple, connected graph with highest degree exactly k so in the case k = 1 we get the classical matching problem. They give a polynomial algorithm, a Tutte-type characterization and a Berge-type minimax formula fo...

The Gallai-Edmonds Decomposition for the k-Piece Packing Problem

An analogue of the Gallai-Edmonds Structure Theorem for non-zero roots of the matching polynomial

Godsil observed the simple fact that the multiplicity of 0 as a root of the matching polynomial of a graph coincides with the classical notion of deficiency. From this fact he asked to what extent classical results in matching theory generalize, replacing “deficiency” with multiplicity of θ as a root of the matching polynomial. We prove an analogue of the Stability Lemma for any given root, whi...

1 Edmonds - Gallai Decomposition and Factor - Critical Graphs

where o(G− U) is the number of connected components in G[V \ U ] with odd cardinality. We call a set U that achieves the minimum on the right hand side of the Tutte-Berge formula, a Tutte-Berge witness set. Such a set U gives some information on the set of maximum matchings in G. In particular we have the following. • All nodes in U are covered in every maximum matching of G. • If K is the vert...

A Gallai-Edmonds-type structure theorem for path-matchings

As a generalization of matchings, Cunningham and Geelen introduced the notion of path-matchings. We give a structure theorem for path-matchings which generalizes the fundamental Gallai-Edmonds structure theorem for matchings. Our proof is purely combinatorial.