Quasi-Transitive Digraphs and Their Extensions

Abstract

A digraph D is quasi-transitive if for any three distinct vertices x, y, z in D, the existence of the arcs xy and yz in D implies that xz, zx or both are arcs of D. Quasi-transitive digraphs generalize both tournaments (and semicomplete digraphs) and transitive digraphs, and share some of the nice properties of these families. In particular, many problems that are \(\mathcal {NP}\)-complete for general digraphs become solvable in polynomial time when restricted to quasi-transitive digraphs. In this chapter, we focus on presenting how usually difficult problems admit efficient solutions for the family of quasi-transitive digraphs and some of its generalizations. We begin with the study of the structure of quasi-transitive digraphs, given by the recursive characterization theorem known as the Canonical Decomposition Theorem; two generalizations of quasi-transitive digraphs are introduced. We define a digraph D to be k-quasi-transitive if for any pair of vertices x, y in D, the existence of a path of length k from x to y implies that xy, yx or both are arcs of D. Given a class of digraphs \(\varPhi \), we say that a digraph is totally \(\varPhi \)-decomposable if it can be expressed as a composition of totally \(\varPhi \)-decomposable digraphs; this concept generalizes the structure of quasi-transitive digraphs given by the Canonical Decomposition Theorem. Some of the problems studied for quasi-transitive digraphs and its generalizations include hamiltonicity, traceability, k-linkages weak k-linkages, existence and number of k-kings, the Path Partition Conjecture and pancyclicity. A brief section is devoted to homomorphisms in transitive digraphs.