We study straight-line drawings of planar graphs with prescribed face areas. A plane graph is area-universal if for every area assignment on the inner faces, there exists a straight-line drawing realizing the prescribed areas. For triangulations with a special vertex order, we present a sufficient criterion for area-universality that only requires the investigation of one area assignment. Moreover, if the sufficient criterion applies to one plane triangulation, then all embeddings of the underlying planar graph are also area-universal. To date, it is open whether area-universality is a property of a plane or planar graph. We use the developed machinery to present area-universal families of triangulations. Among them we characterize area-universality of accordion graphs showing that area-universal and non-area-universal graphs may be structural very similar.
Proposition 2. A plane triangulation T is area-universal if and only if every 4-connected component of T is area-universal.Proof (Sketch). The proof is based on the fact that a plane graph G with a separating triangle t is area-universal if and only if G e , the induced graph by t and its exterior, and G i , the induced graph by t and its interior, are areauniversal. In particular, Lemma 1 allows us to combine realizing drawings of G e and G i to a drawing of G.Remark. Note that a plane 3-tree has no 4-connected component. (Recall that K 4 is 3-connected and a graph on n > 4 vertices is 4-connected if and only if it has no separating triangle.) This is another way to see their area-universality.