In this section we shall be considering graphs from a slightly different point of view, as incidence structures in their own right (as remarked in Chapter 2). In particular, a regular graph is a 1-design with k = 2, and conversely (provided we forbid repeated blocks). An interesting question is: when do such 1-designs have extensions? Since 2-designs with k = 3 are so common, this problem is too general, and we shall usually impose extra conditions on the extension. Extensions of designs were originally used by Witt, Hughes and Dembowski for studying transitive extensions of permutation groups, and it might be expected that extensions of regular graphs would be useful in the study of doubly transitive groups. This is indeed the case.

If we are extending a design by adding an extra point p, we know all the blocks of the extension containing p - these are obtained simply by adjoining p to all the old blocks - but it is in the blocks not containing p that possible ambiguities arise. Let us then suppose, as a first possibility, that we have a set Δ of triples, called blocks, in which the blocks not containing a point p are uniquely determined in a natural way by those containing p; in particular let us suppose that, when we know which of {pqr}, {pqs} and {prs} are blocks, we can decide whether {qrs} is a block.