Bounded embeddings of graphs in the plane

A drawing in the plane (ℝ2) of a graph G = (V,E) equipped with a function γ : V → ℕ is x-bounded if (i) x(u) < x(v) whenever γ(u) < γ(v) and (ii) γ(u) ≤ γ(w) ≤ γ(v), where uv ∈ E and γ(u) ≤ γ(v), whenever x(w) ∈ x(uv), where x(.) denotes the projection to the xaxis.We prove a characterization of isotopy classes of embeddings of connected graphs equipped with γ in the plane containing an x-bounded embedding.Then we present an efficient algorithm, which relies on our result, for testing the existence of an x-bounded embedding if the given graph is a forest.This partially answers a question raised recently by Angelini et al.and Chang et al., and proves that c-planarity testing of flat clustered graphs with three clusters is tractable when the underlying abstract graph is a forest.