A Characterization of Radial Graphs

A level graph G = (V,E, λ) is a graph with a mapping λ : V → {1, . . . , k}, k ≥ 1, that partitions the vertex set V as V = V1 ∪ . . . ∪ Vk, Vj = λ−1(j), Vi ∩ Vj = ∅ for i ̸= j, such that λ(v) = λ(u) + 1 for each edge (u, v) ∈ E. Thus a level planar graph can be drawn with the vertices of every Vj , 1 ≤ j ≤ k, placed on a horizontal line, representing the level lj , and without crossings of edges, which can be drawn as straight line segments between the levels. Healy, Kuusik and Leipert gave a complete characterization of minimal forbidden subgraphs for level planar graphs (MLNP patterns) for hierarchies [4]. Minimal in terms of deleting an arbitrary edge leads to level planarity. A radial graph partitions the vertex set on radii, which can be pictured as concentric circles, instead of levels, lj = (j cos(α), j sin(α)), α ∈ [0, 2π), mapped around a shared center, where j, 1 ≤ j ≤ k indicates the concentric circles’ radius. Comparing embeddings of radial graphs with that of level graphs we gain a further possibility to place an edge and eventually avoid edge crossings which we wish to prevent for planarity reasons. This offers a new set of minimal radial non planar subgraphs (MRNP patterns). Some of the MLNP patterns can be adopted as MRNP patterns while some turn out to be radial planar. But based on the radial planar MLNP patterns and the use of augmentation we can build additional MRNP patterns that did not occur in the level case. Furthermore we point out a new upper bound for the number of edges of radial planar graphs. It depends on the subgraphs induced between two radii. Because of the MRNP patterns these subgraphs can either consist of a forest or a cycle with several branches. Applying the bound we are able to characterize extremal radial planar graphs.