Area-Efficient Drawings of Outer-1-Planar Graphs

We study area-efficient drawings of planar graphs: embeddings of graphs on an integer grid so that the bounding box of the drawing is minimized. Our focus is on the class of outer-1-planar graphs: the family of planar graphs that can be drawn on the plane with all vertices on the outer-face so that each edge is crossed at most once. We first present two straight-line drawing algorithms that yield small-area straight-line drawings for the subclass of complete outer-1-planar graphs. Further, we give an algorithm that produces an orthogonal drawing of any outer-1-plane graph in O(n log n) area while keeping the number of bends per edge relatively small

Lower Bounds on the Area Requirements of Series-Parallel Graphs

Graphs and Algorithm

Lower Bounds on the Area Requirements of Series-Parallel Graphs

We show that there exist series-parallel graphs requiring Ω(n2 √ logn) area in any straightline or poly-line grid drawing. Such a result is achieved in two steps. First, we show that, in any straight-line or poly-line drawing of K2,n, one side of the bounding box has length Ω(n), thus answering two questions posed by Biedl et al. [Information Processing Letters, 2003]. Second, we show a family of series-parallel graphs requiring Ω(2 √ logn) width and Ω(2 √ logn) height in any straight-line or poly-line grid drawing. Combining the two results, the Ω(n2 √ logn) area lower bound is achieved. 2