In [5], Holroyd, Levine, Mészáros, Peres, Propp and Wilson characterize recurrent chip-and-rotor configurations for strongly connected digraphs. However, the number of steps needed to recur, and the number of orbits is left open for general digraphs. Recently, these questions were answered by Pham [6], using linear algebraic methods. We give new, purely combinatorial proofs for these formulas. ...

6 The dicycle transversal number τ(D) of a digraph D is the mini7 mum size of a dicycle transversal of D, i. e. a set T ⊆ V (D) such that 8 D − T is acyclic. We study the following problem: Given a digraph D, 9 decide if there is a dicycle B in D and a cycle C in its underlying undi10 rected graph UG(D) such that V (B)∩V (C) = ∅. It is known that there is 11 a polynomial time algorithm for this...

An in-tournament is an oriented graph such that the in-neighborhood of every vertex induces a tournament. Recently, we have shown that every arc of a strongly connected tournament of order n is contained in a directed path of order r(n + 3)/21. This is no longer valid for strongly connected in-tournaments, because there exist examples containing an arc with the property that the longest directe...

Using the two-way distance, we introduce concept of weak metric dimension a strongly connected digraph [Formula: see text]. We first establish lower and upper bounds for number arcs in text] by using diameter text], characterize all digraphs attaining or bound. Then study with 1 classify vertex-transitive having 1. Finally, order n are determined.

The notion of a C-ultrahomogeneous graph, due to Isaksen et al., is adapted for digraphs, and then a strongly connected ~ C4-ultrahomogeneous oriented graph on 168 vertices and 126 pairwise arc-disjoint 4-cycles is presented, with regular indegree and outdegree 3 and no circuits of lengths 2 and 3, by altering a definition of the Coxeter graph via pencils of ordered lines of the Fano plane in w...

A ~ P3-decomposition of a directed graph D is a partition of the arcs of D into directed paths of length 2. We characterize symmetric digraphs that do not admit a ~ P3-decomposition. We show that the only 2-regular, connected directed graphs that do not admit a ~ P3-decomposition are obtained from undirected odd cycles by replacing each edge by two oppositely directed arcs. In both cases, we gi...

This paper considers a distributed convex optimization problem with inequality constraints over time-varying unbalanced digraphs, where the cost function is a sum of local objectives, and each node of the graph only knows its local objective and inequality constraints. Although there is a vast literature on distributed optimization, most of them require the graph to be balanced, which is quite ...

This technical communique represents a generalization of the convergence analysis for the consensus algorithm proposed in Priolo et al. (2014). Although the consensus was reached for any strongly connected weighted digraphs (SCWD), the convergence analysis provided in Priolo et al. (2014) was only valid for diagonalizable matrices encoding a SCWD. The result we present here generalizes the prev...