Countable locally 2-arc-transitive bipartite graphs

We present an order-theoretic approach to the study of countably infinite locally 2-arc-transitive bipartite graphs. Our approach is motivated by techniques developed by Warren and others during the study of cycle-free partial orders. We give several new families of previously unknown countably infinite locally-2-arc-transitive graphs, each family containing continuum many members. These examples are obtained by gluing together copies of incidence graphs of semilinear spaces, satisfying a certain symmetry property, in a tree-like way. In one case we show how the classification problem for that family relates to the problem of determining a certain family of highly arc-transitive digraphs. Numerous illustrative examples are given.Comment: 29 page

Approximating the Minimum Equivalent Digraph

The MEG (minimum equivalent graph) problem is, given a directed graph, to find a small subset of the edges that maintains all reachability relations between nodes. The problem is NP-hard. This paper gives an approximation algorithm with performance guarantee of pi^2/6 ~ 1.64. The algorithm and its analysis are based on the simple idea of contracting long cycles. (This result is strengthened slightly in ``On strongly connected digraphs with bounded cycle length'' (1996).) The analysis applies directly to 2-Exchange, a simple ``local improvement'' algorithm, showing that its performance guarantee is 1.75.Comment: conference version in ACM-SIAM Symposium on Discrete Algorithms (1994

Highly arc transitive digraphs

Unendliche, hochgradig bogentransitive Digraphen werden definiert und anhand von Beispielen vorgestellt. Die Erreichbarkeitsrelation und Eigenschaft–Z werden definiert und unter Verwendung von Knotengraden, Wachstum und anderen Eigenschaften, die von der Untersuchung von Nachkommen von Doppelstrahlen oder Automorphismengruppen herrühren, auf hochgradig bogentransitiven Digraphen untersucht. Seifters Theoreme über hochgradig bogentransitive Digraphen mit mehr als einem Ende, seine daherrührende Vermutung und deren sie widerlegende Gegenbeispiele werden vorgestellt. Eine Bedingung, unter der C–homogene Digraphen hochgradig bogentransitiv sind, wird angegeben und die Verbindung zwischen hochgradig bogentransitiven Digraphen und total unzusammenhängenden, topologischen Gruppen wird erwähnt. Einige Bemerkungen über die Vermutung von Cameron–Praeger–Wormald werden gemacht und eine verfeinerte Version vermutet. Die Eigenschaften der bekannten hochgradig bogentransitiven Digraphen werden gesammelt. Es wird festgestellt, dass einige, aber nicht alle unter ihnen Cayley–Graphen sind. Schließlich werden offen gebliebene Fragestellungen und Vermutungen zusammengefasst und neue hinzugefügt. Für die vorgestellten Lemmata, Propositionen und Theoreme sind entweder Beweise enthalten, oder Referenzen zu Beweisen werden angegeben.Infinite, highly arc transitive digraphs are defined and examples are given. The Reachability–Relation and Property-Z are defined and investigated on infinite, highly arc transitive digraphs using the valencies, spread and other properties arising from the investigation of the descendants of lines or the automorphism groups. Seifters theorems about highly arc transitive digraphs with more than one end, his conjecture on them and the counterexamples that disproved his conjecture, are given. A condition for C–homogeneous digraphs to be highly arc transitve is stated and the connection between highly arc transitive digraphs and totally disconnected, topological groups is mentioned. Some notes on the Cameron–Praeger–Wormald–Conjecture are made and a refined conjecture is stated. The properties of the known highly arc transitive digraphs are collected, some but not all of them are Cayley–graphs. Finally open questions and conjectures are stated and new ones are added. For the given lemmas, propositions and theorems either proofs or references to proofs are included

Fully Dynamic Single-Source Reachability in Practice: An Experimental Study

Given a directed graph and a source vertex, the fully dynamic single-source reachability problem is to maintain the set of vertices that are reachable from the given vertex, subject to edge deletions and insertions. It is one of the most fundamental problems on graphs and appears directly or indirectly in many and varied applications. While there has been theoretical work on this problem, showing both linear conditional lower bounds for the fully dynamic problem and insertions-only and deletions-only upper bounds beating these conditional lower bounds, there has been no experimental study that compares the performance of fully dynamic reachability algorithms in practice. Previous experimental studies in this area concentrated only on the more general all-pairs reachability or transitive closure problem and did not use real-world dynamic graphs. In this paper, we bridge this gap by empirically studying an extensive set of algorithms for the single-source reachability problem in the fully dynamic setting. In particular, we design several fully dynamic variants of well-known approaches to obtain and maintain reachability information with respect to a distinguished source. Moreover, we extend the existing insertions-only or deletions-only upper bounds into fully dynamic algorithms. Even though the worst-case time per operation of all the fully dynamic algorithms we evaluate is at least linear in the number of edges in the graph (as is to be expected given the conditional lower bounds) we show in our extensive experimental evaluation that their performance differs greatly, both on generated as well as on real-world instances