Linear-Time Rectilinear Drawings of Subdivisions of Triconnected Cubic Planar Graphs with Orthogonally Convex Faces

A graph is called planar if it admits a planar drawing on the plane, i.e., no two edges create a crossing except possibly at their common endpoint. In a rectilinear drawing I' of a planar graph, each vertex is drawn as a point and each edge is drawn as either horizontal or vertical line segment. A face in I' is called orthogonally convex if every horizontal or vertical line segment connecting two points within the face does not intersect any other face. We examine the decision problem that takes a planar graph as an input and seeks for a rectilinear drawing where the faces are drawn as orthogonally convex polygons. A linear-time algorithm for this problem is known for biconnected planar graphs, but the algorithm relies on complex data structures and linear-time planarity testing, which are challenging to implement. In this paper, we give a necessary and sufficient condition for a subdivision of a triconnected cubic planar graph to admit such a drawing, and design a linear-time algorithm to check the condition and compute a desired drawing, if it exists. As a byproduct of our results we show that if a subdivision of a triconnected cubic planar graph G admits a rectilinear drawing, then it must also admit a rectilinear drawing with orthogonally convex faces.