Semi-transitive graphs

In this paper, we first consider graphs allowing symmetry groups which act transitively on edges but not on darts (directed edges). We see that there are two ways in which this can happen and we introduce the terms bi-transitive and semi-transitive to describe them. We examine the elementary implications of each condition and consider families of examples; primary among these are the semi-transitive spider-graphs PS(k,N;r) and MPS(k,N;r). We show how a product operation can be used to produce larger graphs of each type from smaller ones. We introduce the alternet of a directed graph. This links the two conditions, for each alternet of a semi-transitive graph (if it has more than one) is a bi-transitive graph. We show how the alternets can be used to understand the structure of a semi-transitive graph, and that the action of the group on the set of alternets can be an interesting structure in its own right. We use alternets to define the attachment number of the graph, and the important special cases of tightly attached and loosely attached graphs. In the case of tightly attached graphs, we show an addressing scheme to describe the graph with coordinates. Finally, we use the addressing scheme to complete the classification of tightly attached semi-transitive graphs of degree 4 begun by Marušič and Praeger. This classification shows that nearly all such graphs are spider-graphs.