In this paper we address the edge-agreement problem with preserved connectivity for networks of first and second-order systems under proximity constraints interconnected over a class directed graphs. We provide strict Lyapunov function that leads to establishing uniform asymptotic stability consensus manifold guaranteed preservation. Furthermore, robustness protocol, in sense input-to-state res...

An acyclic set in a digraph is a set of vertices that induces an acyclic subgraph. In 2011, Harutyunyan conjectured that every planar digraph on n vertices without directed 2-cycles possesses an acyclic set of size at least 3n/5. We prove this conjecture for digraphs where every directed cycle has length at least 8. More generally, if g is the length of the shortest directed cycle, we show that...

Henning and Yeo [SIAM J. Discrete Math. 26 (2012) 687–694] conjectured that a 3-regular digraph D contains two vertex disjoint directed cycles of different length if either D is of sufficiently large order or D is bipartite. In this paper, we disprove the first conjecture. Further, we give support for the second conjecture by proving that every bipartite 3-regular digraph, which either possesse...

Given a weighted directed graph G = (V,A), the minimum feedback arc set problem consists of finding a minimum weight set of arcs A ⊆ A such that the directed graph (V,A \ A) is acyclic. Similarly, the minimum feedback vertex set problem consists of finding a minimum weight set of vertices containing at least one vertex for each directed cycle. Both problems are NP-complete. We present simple co...

We call the digraph D an m-coloured digraph if the arcs of D are coloured with m colours. A directed path (or a directed cycle) is called monochromatic if all of its arcs are coloured alike. A set N ⊆ V (D) is said to be a kernel by monochromatic paths if it satis5es the following two conditions: (i) For every pair of di7erent vertices u, v∈N , there is no monochromatic directed path between th...

Given a weighted directed hypergraph H = (V,EH ;w), where w : EH → R+, we consider the problem of embedding all weighted directed hyperedges on a mixed cycle, which consists of undirected and directed links. The objective is to minimize the maximum congestion of any undirected or directed link in the mixed cycle. In this paper, we first formulate this new problem as an integer linear program, a...

Using a recent result of Chudnovsky, Seymour, and Sullivan, we slightly improve two bounds related to the Caccetta-Haggkvist Conjecture. Namely, we show that if α ≥ 0.35312, then each n-vertex digraph D with minimum outdegree at least αn has a directed 3-cycle. If β ≥ 0.34564, then every n-vertex digraph D in which the outdegree and the indegree of each vertex is at least βn has a directed 3-cy...

Let G be an Eulerian bipartite digraph with vertex partition sizes m,n. We prove the following Turán-type result: If e(G) > 2mn/3 then G contains a directed cycle of length at most 4. The result is sharp. We also show that if e(G) = 2mn/3 and no directed cycle of length at most 4 exists, then G must be biregular. We apply this result in order to obtain an improved upper bound for the diameter o...

The main result in this paper is that for m == 0 (mod 6) a partial directed m-cycle system of order n can be embedded in a directed m-cycle system of order less than (mn)/2 + m/2 + 2m + 1. For fixed m, this bound is asymptotic in n to (mn) /2 which is approximately one-half of the best known bound of mn + (0 or 1).