In this note we show that the concept of incidence coloring introduced in [BM] is a special case of directed star arboricity, introduced in [AA]. A conjecture in [BM] concerning asmyptotics of the incidence coloring number is solved in the negative following an example in [AA]. We generalize Theorem 2.1 of [AMR] concerning the star arboricity of graphs to the directed case by a slight modificat...

Given a simple directed graph D = (V,A), let the size of the largest induced directed acyclic graph (dag) be denoted by mas(D). Let D ∈ D(n, p) be a random instance, obtained by choosing each of the ( n 2 ) possible undirected edges independently with probability 2p and then orienting each chosen edge independently in one of two possible directions with probabibility 1/2. We obtain improved bou...

Gorodezky and Pak (Random Struct. Algorithms, 2014) introduced a “clusterpopping” algorithm for sampling root-connected subgraphs in a directed graph, and conjectured that it runs in expected polynomial time on bi-directed graphs. We confirm their conjecture. It follows that there is a fully polynomial-time randomized approximation scheme (FPRAS) for reachability in bi-directed graphs. Reachabi...

Catalan numbers arise in a family of persymmetric arrays with determinant 1. The demonstration involves a counting result for disjoint path systems in acyclic directed graphs.

An asteroidal triple in a graph is a set of three non-adjacent vertices such that for any two of them there exists a path between them that does not intersect the neighborhood of the third. An asteroidal quadruple is a set of four non-adjacent vertices such that any three of them is an asteroidal triple. In this paper, we study a subclass of directed path graph, the class of extended star direc...

The study of distance-regular directed graphs can be reduced to that of short distance-regular directed graphs. We consider the eigenspaces of the intersection matrix of a short distance-regular directed graph and show that nearly all the eigenvalues are nonreal. Next we show that a nontrivial short distance-regular directed graph is primitive.

Visualizing information whose underlying graph is directed acyclic is easier if the graph is Minimal Vertex Series Parallel (MVSP). We present method of transforming any directed acyclic graph into one that is MVSP. This enables an easy visualization of the information contained in the original graph. We illustrate this by an example of a schedule that contains parallelism.

Although there are many existing alternative methods for using structural characterizations of undirected graphs for embedding, clustering and classification problems, there is relatively little literature aimed at dealing with such problems for directed graphs. In this paper we present a novel method for characterizing graph structure that can be used to embed directed graphs into a feature sp...

Given a graph, does there exist an orientation of the edges such that the resulting directed graph is strongly connected? Robbins’ theorem [Robbins, Am. Math. Monthly, 1939] states that such an orientation exists if and only if the graph is 2-edge connected. A natural extension of this problem is the following: Suppose that the edges of the graph is partitioned into trails. Can we orient the tr...