Book Chapter10.1007/978-3-642-02029-2_18
Path Partitions, Cycle Covers and Integer Decomposition
András Sebő
- 27 Jul 2009
- pp 183-199
TL;DR: A polyhedron P has the integer decomposition property if every integer vector in kP is the sum of k integer vectors in P. In as mentioned in this paper, it was shown that the projections of polyhedra defined by totally unimodular constraint matrices have the same property.
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Abstract: A polyhedron P has the integer decomposition property , if every integer vector in kP is the sum of k integer vectors in P . We explain that the projections of polyhedra defined by totally unimodular constraint matrices have the integer decomposition property, in order to deduce the same property for coflow polyhedra defined by Cameron and Edmonds. We then apply this result to the convex hull of particular stable sets in graphs. Therebye we prove a generalization of Greene and Kleitman's well-known theorem on posets to arbitrary digraphs which implies recent and classical purely graph theoretical results on cycle covers, is closely related to conjectures of Berge and Linial on path partitions, and implies these for some particular values of the parameters.
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Citations
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William J. Cook,William H. Cunningham,William R. Pulleyblank,Alexander Schrijver +3 more
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2.9K
The structure of sperner k-families
Curtis Greene,Daniel J. Kleitman +1 more
TL;DR: An extension of Dilworth's theorem is obtained by relating the maximum size of a k-family to certain partitions of P into chains, and a natural lattice ordering on k-families is defined and analyzed.
229
Comparability graphs and intersection graphs
TL;DR: A function diagram (f-diagram) D consists of the family of curves obtained from n continuous functions f"i:[0,1]->R(1=1) and on the other side of the diagonal R is the graph of the integers.
164
Intersection graphs of paths in a tree
TL;DR: A unified framework for studying several classes of intersection graphs arising from families of paths in a tree is presented, which clarifies and unifies virtually all of the important known results for these graphs and provides substantial new results as well.
158