We consider directed graph algorithms in a streaming setting, focusing on problems concerning orderings of the vertices. This includes such fundamental problems as topological sorting and acyclicity testing. We also study the related problems of finding a minimum feedback arc set (edges whose removal yields an acyclic graph), and finding a sink vertex. We are interested in both adversarially-ordered and randomlyordered streams. For arbitrary input graphs with edges ordered adversarially, we show that most of these problems have high space complexity, precluding sublinear-space solutions. Some lower bounds also apply when the stream is randomly ordered: e.g., in our most technical result we show that testing acyclicity in the p-pass random-order model requires roughly n 1+1/p space. For other problems, random ordering can make a dramatic difference: e.g., it is possible to find a sink in an acyclic tournament in the one-pass randomorder model using polylog(n) space whereas under adversarial ordering roughly n 1/p space is necessary and sufficient given Θ(p) passes. We also design sublinear algorithms for the feedback arc set problem in tournament graphs; for random graphs; and for randomly ordered streams. In some cases, we give lower bounds establishing that our algorithms are essentially space-optimal. Together, our results complement the much maturer body of work on algorithms for undirected graph streams.
In this paper we study oriented bipartite graphs. In particular, we introduce bitransitive graphs and bitournaments. Several characterizations of bitransitive bitournaments are obtained. Next we prove the Caccetta-Häggkvist Conjecture for oriented bipartite graphs for some cases for which it is unsolved in general. We introduce the concept of oriented odd-even graphs and (undirected) odd-even graphs and characterize (oriented) bipartite graphs in terms of them.In fact, we show that any (oriented) bipartite graph can be represented by some (oriented) odd-even graph. We obtain some conditions for connectedness of odd-even graphs. Finally we introduce Goldbach graphs which are special type of odd-even graphs. We show that the famous Goldbach's conjecture is equivalent to the connectedness of Goldbach graphs. Several other related conjectures are related to various parameters of Goldbach graphs. We study nature of degrees of vertices and independent sets of Goldbach graphs.
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