The EdmondsGallai Decomposition for the kPiece Packing Problem Generalizing Kanekou0027s long path packing problem, Hartvigsen, Hell and Szabo consider a new type of undirected graph packing problem, called the  kpiece packing problem . A kpiece is a simple, connected graph with highest degree exactly k so in the case k1 we get the classical matching problem. They give a polynomial algorithm, a Tuttetype characterization and a Bergetype minimax formula for the kpiece packing problem. However, they leave open the question of an EdmondsGallai type decomposition. This paper fills this gap by describing such a decomposition. We also prove that the vertex sets coverable by kpiece packings have a certain matroidal structure.

Packing and Covering with Segments We study three fundamental geometric optimization problems xe2x80x93 independent set, piercing set, and dominating set xe2x80x93 on sets of axisparallel segments in the plane. We consider special cases in which the segments are either unit length or they are anchored on an inclined line (a line with slope (1)). When the segments are anchored on both sides, we prove that all three problems are NPcomplete (Throughout, we refer to NPcompleteness of problems, as the decision versions of the NPhard optimization problems we consider are all readily seen to be in NP.); NPcompleteness was known for the corresponding problems with axisparallel rectangles anchored on an inclined line (Correa et al. 4, Mudgal and Pandit 9, Pandit 10). Further, we prove that the dominating set problem with unit segments in the plane is NPcomplete. When the input segments are anchored on one side of the inclined line, there are polynomialtime algorithms for the independent set and piercing set problems.