We consider the problem of orienting the edges of a planar graph in such a way that the out-degree of each vertex is minimized. If, for each vertex v, the out-degree is at most d, then we say that such an orientation is d-bounded. We prove the following results: ffl Each planar graph has a 5-bounded acyclic orientation, which can be constructed in linear time. ffl Each planar graph has a 3-bounded orientation, which can be constructed in linear time. ffl A 6-bounded acyclic orientation, and a 3-bounded orientation, of each planar graph can each be constructed in parallel time O(log n log n) on an EREW PRAM, using O(n= log n log n) processors. As an application of these results, we present a data structure such that each entry in the adjacency matrix of a planar graph can be looked up in constant time. The data structure uses linear storage, and can be constructed in linear time. Department of Mathematics and Computer Science, University of California, Riverside, CA 92521. On...

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 and combinatorial optimization.

The network design problem with connectivity requirements (NDC) models a wide variety of celebrated combinatorial optimization problems including the minimum spanning tree, Steiner tree, and survivable network design problems. We develop strong formulations for two versions of the edge-connectivity NDC problem: unitary problems re-quiring connected network designs, and nonunitary problems permitting non-connected networks as solutions. We (i) present a new directed formulation for the unitary NDC problem that is stronger than a natural undirected formulation, (ii) project out several classes of valid inequalities-partition inequalities, odd-hole inequalities, and combinatorial design inequalities-that generalize known classes of valid inequalities for the Steiner tree problem to the unitary NDC problem, and (iii) show how to strengthen and direct nonunitary problems. Our results provide a unifying framework for strengthening formulations for NDC problems, and demonstrate the strength and power of flow-based formulations for net-work design problems with connectivity requirements.

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We give a constructive characterization of undirected graphs which contain k spanning trees after adding any new edge. This is a generalization of a theorem of Henneberg and Laman who gave the characterization for k = 2. We also give a constructive characterization of graphs which have k edge-disjoint spanning trees after deleting any edge of them.

Parity (matching theory) and connectivity (network flows) are two main branches of combinatorial optimization. In an attempt to understand better their interrelation, we study a problem where both parity and connectivity requirements are imposed. The main result is a characterization of undirected graphs G = (V; E) having a k-edge-connected T -odd orientation for every subset T ` V with jEj + jT j even. (T -odd orientation: the in-degree of v is odd precisely if v is in T .) As a corollary, we obtain that every (2k + 2)-edge-connected graph with jV j + jEj even has a k-edge-connected orientation in which the in-degree of every node is odd. Along the way, a structural characterization will be given for digraphs with a root-node s having k + 1 edge-disjoint paths from s to every node and k edge-disjoint paths from every node to s. 1. INTRODUCTION The notion of parity plays an important role in describing combinatorial structures. The prime example is W. Tutte's theorem [T47] on the exi...

Generalizing an earlier result of H.E. Robbins [1939], C.St.J.A. Nash-Williams [1960] proved that an undirected graph G has a k-edge-connected orientation if and only if G is 2k-edge-connected. In a recent paper Nash-Williams [1995] found a necessary and sufficient condition for the existence of a strongly-connected orientation of a mixed graph so that every node v has at least a prescribed number of newly oriented edges entering v. It was known earlier how the first of these theorems derives from the theory of submodular flows. In this paper we describe how (a generaliztion of) the second does. As a main device, we prove a simplified feasibility theorem for submodular flows constrained by crossing submodular functions. I. INTRODUCTION Let G = (V; E) be an undirected graph and h : 2 V ! Z [ f\Gamma1g an integer-valued set-function with h(;) = h(V ) = 0. The general form of the orientation problem we consider consists of finding an orientation of the edges of G so that in the result...

We study the problem of orienting the edges of a weighted graph such that the maximum weighted outdegree of vertices is minimized. This problem, which has applications in the guard arrangement for example, can be shown to be generally. In this paper we first give optimal orientation algorithms which run in polynomial time for the following special cases: (i) the input is an unweighted graph, or more generally, a graph with identically weighted edges, and (ii) the input graph is a tree. Then, by using those algorithms as sub-procedures, we provide a simple, combinatorial, min{ , (2-#)}-approximation algorithm for the general case, where wmax and w min are the maximum and the minimum weights of edges, respectively, and # is some small positive real number that depends on the input.

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Given an undirected hypergraph and a subset of vertices S ⊆ V with a specified root vertex r ∈ S, the Steiner Rooted-Orientation problem is to find an orientation of all the hyperedges so that in the resulting directed hypergraph the “connectivity” from the root r to the vertices in S is maximized. This is motivated by a multicasting problem in undirected networks as well as a generalization of some classical problems in graph theory. The main results of this paper are the following approximate min-max relations: • Given an undirected hypergraph H, if S is 2k-hyperedge-connected in H, then H has a Steiner rooted k-hyperarc-connected orientation. • Given an undirected graph G, if S is 2k-element-connected in G, then G has a Steiner rooted k-element-connected orientation. Both results are tight in terms of the connectivity bounds. These also give polynomial time constant factor approximation algorithms for both problems. The proofs are based on submodular techniques, and a graph decomposition technique used in the Steiner Tree Packing problem. Some complementary hardness results are presented at the end.