A \(G\)-design of order \(n\) is a pair \((P, B)\) where \(P\) is the vertex set of \(K_n\), the complete graph on \(n\) points, and \(B\) is an edge- disjoint decomposition of \(K_n\) into isomorphic copies of the simple graph \(G\) (and such copies are called blocks). Given a graph \(G\), the intersection problem asks for which \(k\) there exist two \(G\)-designs \((P, B_1)\) and \((P, B_2)\) of order \(n\) such that \(|B_1 \cap B_2 |= k\).
The authors solve the intersection problem for the following graphs: paths of length 2, 3, and 4 (the length being the number of edges), stars with 3 and 4 edges, a triangle with a pendant edge, a tree with five vertices having one vertex of degree two and one vertex of degree three.
This paper completes the already known results on the intersection problem for small graphs, so that this problem is solved for all connected graphs \(G\) with at most four vertices or at most four edges.