2-placement of (p, Q)-trees

Let G = (L,R; E) be a bipartite graph such that V (G) = L ∪ R, |L| = p and |R| = q. G is called (p, q)-tree if G is connected and |E(G)| = p + q − 1. Let G = (L, R;E) and H = (L′, R′;E′) be two (p, q)-tree. A bijection f : L ∪ R → L′ ∪ R′ is said to be a biplacement of G and H if f(L) = L′ and f(x)f(y) / ∈ E′ for every edge xy of G. A biplacement of G and its copy is called 2-placement of G. A bipartite graph G is 2-placeable if G has a 2-placement. In this paper we give all (p, q)-trees which are not 2-placeable.