On the Structure of Local Tournaments

Abstract

A local tournament is an oriented graph in which the inset, as well as the outset, of every vertex induces a tournament. Local tournaments are interesting in their own right, as they share many nice properties of tournaments. They are also of interest because of their relation to proper circular are graphs. Indeed, a connected graph can be oriented as a local tournament if and only if it is a proper circular are graph. Thus understanding the structure of local tournaments also sheds light on the structure of proper circular are graphs. We describe a method to generate all local tournaments by performing some simple operations on some simple basic oriented graphs. As a consequence we obtain a description of all local tournaments with the same underlying proper circular are graph. Similar structural information is obtained for non-strong local tournaments, which are related to proper interval graphs. These structure theorems have already been used to design optimal algorithms to recognize proper circular are graphs, proper interval graphs, and local tournaments (and to represent them if possible) and to find a maximum clique in a represented proper circular are graphs.