Abstract. For positive integers m and n with 1 ≤ m ≤ n, the m-competition index (generalized competition index) of a primitive digraph D of order n is the smallest positive integer k such that for every pair of vertices x and y in D, there exist m distinct vertices v1, v2, . . . , vm such that there exist walks of length k from x to vi and from y to vi for each i = 1, . . . , m. In this paper, ...

Neighborhood specification is a dominant consideration in assuring the success of a direct search approach to a difficult combinatorial optimization problem. Previous research has shown the efficacy of imposing an elementary landscape upon the search topology. Barnes et al. [J.W. Barnes, S. Dokov, B. Dimova, A. Solomon, A theory of elementary landscapes, Applied Mathematics Letters 16 (2003)] g...

In a Cayley digraph on a group G, if a distinct colour is assigned to each arc-orbit under the left-regular action of G, it is not hard to show that the elements of the left-regular action of G are the only digraph automorphisms that preserve this colouring. In this paper, we show that the equivalent statement is not true in the most straightforward generalisation to G-vertex-transitive digraph...

The descendant set desc(α) of a vertex α in a digraph D is the set of vertices which can be reached by a directed path from α. A subdigraph of D is finitely generated if it is the union of finitely many descendant sets and D is descendant-homogeneous if it is vertex transitive and any isomorphism between finitely generated subdigraphs extends to an automorphism. We consider connected descendant...

A simple digraph describes the off-diagonal zero-nonzero pattern of a family of (not necessarily symmetric) matrices. Minimum rank of a simple digraph is the minimum rank of this family of matrices; maximum nullity is defined analogously. The simple digraph zero forcing number is an upper bound for maximum nullity. Cut-vertex reduction formulas for minimum rank and zero forcing number for simpl...

We present an order-theoretic approach to the study of countably infinite locally 2-arc-transitive bipartite graphs. Our approach is motivated by techniques developed by Warren and others during the study of cycle-free partial orders. We give several new families of previously unknown countably infinite locally-2-arc-transitive graphs, each family containing continuum many members. These exampl...

We construct a family of infinite, non-locally finite highly arc-transitive digraphs which do not have universal reachability relation and which omit special digraphs called ‘crowns’. Moreover, there is no homomorphism from any of our digraphs onto Z. The methods are adapted from [5] and [6].

Let D be a locally finite, connected, 1-arc transitive digraph. It is shown that the reachability relation is not universal in D provided that the stabilizer of an edge satisfies certain conditions which seem to be typical for highly arc transitive digraphs. As an implication, the reachability relation cannot be universal in highly arc transitive digraphs with prime inor out-degree. Two differe...

Transitive tournament (including transitive triangle) and its blow-up have some symmetric properties. In this work, we establish an analogue result of the Erdös-Stone theorem weighted digraphs with a forbidden tournament. We give stability oriented graphs triangles show that almost all are bipartite, which reconfirms strengthens conjecture Cherlin.