Book contents
- Frontmatter
- Contents
- Preface
- Restricted colorings of graphs
- Polynomials in finite geometries and combinatorics
- Models of random partial orders
- Applications of submodular functions
- Weighted quasigroups
- Graphs with projective subconstituents which contain short cycles
- On circuit covers, circuit decompositions and Euler tours of graphs
- Slicing the hypercube
- Combinatorial designs and cryptography
Applications of submodular functions
Published online by Cambridge University Press: 16 March 2010
- Frontmatter
- Contents
- Preface
- Restricted colorings of graphs
- Polynomials in finite geometries and combinatorics
- Models of random partial orders
- Applications of submodular functions
- Weighted quasigroups
- Graphs with projective subconstituents which contain short cycles
- On circuit covers, circuit decompositions and Euler tours of graphs
- Slicing the hypercube
- Combinatorial designs and cryptography
Summary
ABSTRACT Submodular functions and related polyhedra play an increasing role in combinatorial optimization. The present survey-type paper is intended to provide a brief account of this theory along with several applications in graph theory.
INTRODUCTION
In 1960 C.St.J.A. Nash-Williams generalized the following easy but pretty result of H.E. Robbins [1939]: the edges of an undirected graph G can be oriented so that the resulting directed graph D is strongly connected if and only if G is 2-edge-connected.
To formulate the generalization let us call a directed graph [undirected graph] κ-edge-connected if there are κ edge-disjoint directed (undirected)) paths from each node to each other.)
WEAK ORIENTATION THEOREM 1.1 [Nash-Williams, 1960] The edges of an undirected graph G can be oriented so that the resulting directed graph is κ-edge-connected if and only if G is 2κ-edge-connected.
The neccessity of the condition is straightforward and the main difficulty lies in proving its sufficiency. Actually, Nash-Williams proved a stronger result. To formulate it, we need the following notation. Given a directed or undirected graph G, let λ(x, y; G) denote local edge-connectivity from x to y, that is, the maximum number of edge-disjoint paths from x to y.
- Type
- Chapter
- Information
- Surveys in Combinatorics, 1993 , pp. 85 - 136Publisher: Cambridge University PressPrint publication year: 1993
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