Arc and twin arc domination in Cayley digraphs

Restricted domination in arc-colored digraphs

Let H = (V (H), A(H)) be a digraph possibly with loops and D = (V (D), A(D)) a digraph whose arcs are colored with the vertices of H (this is what we call an H-colored digraph); i.e. there exists a function c : A(D) → V (H); for an arc of D, f = (u, v) ∈ A(D), we call c(f) = c(u, v) the color of f . A directed walk (directed path) P = (u0, u1, . . . , un) in D will be called an H-walk (H-path) ...

Super-connected arc-transitive digraphs

A digraph is said to be super-connected if every minimum vertex cut is the out-neighbor set or in-neighbor set of a vertex. A digraph is said to be reducible, if there are two vertices with the same out-neighbor set or the same in-neighbor set. In this paper, we prove that a strongly connected arc-transitive oriented graph is either reducible or super-connected. Furthermore, if this digraph is ...

Bi-Arc Digraphs and Conservative Polymorphisms

We introduce the class of bi-arc digraphs, and show they coincide with the class of digraphs that admit a conservative semi-lattice polymorphism, i.e., a min ordering. Surprisingly this turns out to be also the class of digraphs that admit totally symmetric conservative polymorphisms of all arities. We give an obstruction characterization of, and a polynomial time recognition algorithm for, thi...

Arc-Disjoint Paths in Expander Digraphs

Highly arc-transitive digraphs – counterexamples and structure∗

We resolve two problems of [Cameron, Praeger, and Wormald – Infinite highly arc transitive digraphs and universal covering digraphs, Combinatorica 1993]. First, we construct a locally finite highly arc-transitive digraph with universal reachability relation. Second, we provide constructions of 2-ended highly arc transitive digraphs where each ‘building block’ is a finite bipartite graph that is...