Let S be a set and F = {S1, S2, . . . , Sp} be a non-empty family of distinct nonempty subsets of S whose union is S. The intersection graph of F is denoted by I(F) and defined by V (I(F)) = F, with Si and Sj adjacent whenever i not equal to j and Si ∩ Sj not equal to ∅. Then a graph G is an intersection graph on S if there exists a family F of subsets for which G and I (F) are isomorphic graphs. The intersection number omega(G) of a given graph G is the minimum number of elements in a set S such that G is an intersection graph on S.

If G is a connected (p, q)-graph and p ≥ 3 then omega(G) ≤ q. Let G be a connected (p, q)-graph with p > 3. Then omega(G) = q if and only if G has no triangles.

If H is an induced subgraph of a graph G; then omegaG) ≥ omega(H).