Linear-Time Rectilinear Drawings of Subdivisions of Triconnected Cubic Planar Graphs with Orthogonally Convex Faces
Discrete and Computational Geometry, Graphs, and Games
Keywords:graph drawing, rectilinear drawing, orthogonally convex face, subdivision
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 $\Gamma$ 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 $\Gamma$ 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.