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The distinguishing number D(G) of a graph G is the least cardinal number ℵ such that G has a labeling with ℵ labels that is only preserved by the trivial automorphism. We show that the distinguishing number of the countable random graph is two, that tree-like graphs with not more than continuum many vertices have distinguishing number two, and determine the… (More)

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… (More)

The growth function of a graph with respect to a vertex is near polynomial if there exists a polynomial bounding it above for infinitely many positive integers. In the paper vertex-symmetric undirected graphs and vertex-symmetric directed graphs with coinciding in-and out-degrees are described in the case their growth functions are near polynomial. 1. Let Γ… (More)

The class of all connected vertex-transitive graphs forms a metric space under a natural combinatorially defined metric. In this paper we study graphs which are limit points in this metric space of the subset consisting of all finite graphs that admit a vertex-primitive group of automorphisms. A description of these limit graphs provides a useful… (More)

In this paper we investigate reachability relations on the vertices of digraphs. If W is a walk in a digraph D, then the height of W is equal to the number of edges traversed in the direction coinciding with their orientation, minus the number of edges traversed opposite to their orientation. Two vertices u, v ∈ V (D) are R a,b-related if there exists a… (More)