Berge’s conjecture from 1982 on path partitions in directed graphs generalizes and extends Dilworth’s Theorem and the Greene-Kleitman Theorem which are well known for partially ordered sets. The conjecture relates path partitions to a collection of k independent sets, for each k ≥ 1. The conjecture is still open and intriguing for all k > 1. In this paper, we will survey partial results on the ...

Let D be a directed graph of order n. An anti-directed (hamiltonian) cycleH inD is a (hamiltonian) cycle in the graph underlyingD such that no pair of consecutive arcs in H form a directed path in D. In this paper we give sufficient conditions for the existence of anti-directed hamiltonian cycles. Specifically, we prove that a directed graph D of even order n with minimum indegree and outdegree...

An asteroidal triple is a stable set of three vertices such that each pair is connected by a path avoiding the neighborhood of the third vertex. Asteroidal triples play a central role in a classical characterization of interval graphs by Lekkerkerker and Boland. Their result says that a chordal graph is an interval graph if and only if it contains no asteroidal triple. In this paper, we prove a...

The chordality of an undirected graph G, which is not acyclic, is defined as the length of the longest induced cycle in it. The chordality of an acyclic graph is defined to be 0. We use Cl (l ≥ 3) to denote a cycle of length l. An induced cycle is called a hole. A hole is an odd hole if its length is odd and is an even hole otherwise. Odd-chordality of a graph is the length of the longest odd h...

Given an n-vertex planar directed graph with real edge lengths and with no negative cycles, we show how to compute single-source shortest path distances in the graph in O(n log n/ log log n) time with O(n) space. This improves on a recent O(n log n) time bound by Klein et al.

It is well-known that in a directed graph, if deleting any edge will not affect the shortest distance between two specific vertices s and t, then there are two edge-disjoint paths from s to t and both of them are shortest paths. In this paper, we generalize this to shortest k edge-disjoint s-t paths for any positive integer k.