The concept of a line digraph is generalized to that of a directed path graph. The directed path graph → Pk(D) of a digraph D is obtained by representing the directed paths on k vertices of D by vertices. Two vertices are joined by an arc whenever the corresponding directed paths in D form a directed path on k + 1 vertices or form a directed cycle on k vertices in D. In this introductory paper ...

Many polynomial invariants are defined on graphs for encoding the combinatorial information and researching them algebraically. In this paper, we introduce the cycle polynomial and the path polynomial of directed graphs for counting cycles and paths, respectively. They satisfy recurrence relations with respect to elementary edge or vertex operations. They are related to other polynomials and ca...

A digraph is odd-cycle-symmetric if every arc in any elementary odd directed cycle has the reverse arc. This concept arises in the context of the even factor problems, which generalize the path-matching problems. While the even factor problem is NP-hard in general digraphs, it is solvable in polynomial time for odd-cycle-symmetric digraphs. This paper provides a characterization of odd-cycle-sy...

A cycle cover of a graph is a set of cycles such that every vertex is part of exactly one cycle. An L-cycle cover is a cycle cover in which the length of every cycle is in the set L ⊆ N. For most sets L, the problem of computing L-cycle covers of maximum weight is NP-hard and APX-hard. We devise polynomial-time approximation algorithms for L-cycle covers. More precisely, we present a factor 2 a...

Preface Tournaments constitute perhaps the most well-studied class of directed graphs. One of the reasons for the interest in the theory of tournaments is the monograph Topics on Tournaments [58] by Moon published in 1968, covering all results on tournaments known up to this time. In particular, three results deserve special mention: in 1934 Rédei [60] proved that every tournament has a directe...

The present paper investigates the robustness of the consensus protocol over weighted directed graphs using the Nyquist criterion. The limit to which a single weight can vary, while consensus among the agents can be achieved, is explicitly derived. It is shown that even with a negative weight on one of the edges, consensus may be achieved. The result obtained in this paper is applied to a direc...

In this paper we present a distributed algorithm for detecting cycles in large-scale directed graphs, along with its correctness proof and analysis. The algorithm is then extended to find strong components in directed graphs. We indicate an application to detecting cycles in number theoretic functions such as the proper divisor function. Our prototype implementation of the cycle detection algor...

Proteins are directed to cellular compartments by specific localization signals. A GTP-driven cycle has now been identified as a mechanism for protein targeting to the nucleolus. The involvement of a GTP switch suggests that nucleolar localization can be regulated and may be responsive to extracellular stimuli via signaling pathways. The uncovered mechanism also implies that localization is det...

We prove the following approximate version of Pósa’s theorem for directed graphs: every directed graph on n vertices whose inand outdegree sequences satisfy di ≥ i+o(n) and d+i ≥ i+o(n) for all i ≤ n/2 has a Hamilton cycle. In fact, we prove that such digraphs are pancyclic (i.e. contain cycles of lengths 2, . . . , n). We also prove an approximate version of Chvátal’s theorem for digraphs. Thi...