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Parameters of Partial Orders and Graphs: Packing, Covering, and Representation

  • Chapter
Graphs and Order

Part of the book series: NATO ASI Series ((ASIC,volume 147))

Abstract

This paper surveys results concerning packing/covering parameters and representation parameters of graphs and posets. We start from the poset param¬eters known as width and dimension. We consider generalizations, related ques¬tions, and analogous parameters for graphs and/or directed graphs.

Dilworth’s Theorem states that a poset with finite width (maximum antichain size) w can be covered with w chains; we review the known proofs. Greene and Kleitman generalized this to unions of k antichains, called k-famities, proving that the maximum size of a k-family equals the minimum in a dual chain-covering problem. Chain coverings yielding the optimal bound on k-family size are k-saturated partitions. We review proofs and discuss other aspects of the theory of saturated partitions. For example, chain coverings become cliques in the graph context or path coverings in the context of directed graphs, and duality questions still apply. Other topics include duality questions for product orders or product graphs, and the study of element sets that meet all maximal chains in a poset or maximal cliques in a graph.

Packing and covering focus on vertex subsets; “representation” expresses the entire relation as the union or intersection of a minimum number of “nice” relations. We review results on order dimension (the minimum number of chains whose intersection is the poset) and study various dimension parameters for graphs and digraphs. The Ferrers dimension of a digraph D is the minimum number of “Ferrers digraphs” whose intersection is D. Bouchet and Cogis proved that the Ferrers dimension of a reflexive, transitive, antisymmetric relation equals its order dimension as a poset. For undirected graphs, we discuss threshold dimension, product dimension, and boxicity. In each case we seek results analogous to those for order dimension of posets. The last of these leads to a general discussion of intersection representations of graphs, multiple inter¬section parameters, and edge-covering problems.

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  1. Mathematics Department, Univ. of Illinois, Urbana, Illinois, 61801, USA

    Douglas B. West

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  1. Department of Mathematics and Statistics, The University of Calgary, Calgary, Alberta, Canada

    Ivan Rival

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West, D.B. (1985). Parameters of Partial Orders and Graphs: Packing, Covering, and Representation. In: Rival, I. (eds) Graphs and Order. NATO ASI Series, vol 147. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-5315-4_8

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