Abstract
A non-aligned drawing of a graph is a drawing where no two vertices are in the same row or column. Auber et al. showed that not all planar graphs have non-aligned drawings that are straight-line, planar, and in the minimal-possible $n\times n$-grid. They also showed that such drawings exist if up to $n-3$ edges may have a bend. In this paper, we give algorithms for non-aligned planar drawings that improve on the results by Auber et al. In particular, we give such drawings in an $n\times n$-grid with significantly fewer bends, and we study what grid-size can be achieved if we insist on having straight-line drawings.
Highlights
At the GD 2015 conference, Auber et al [3] introduced the concept of rookdrawings: These are drawings of a graph in an n × n-grid such that no two vertices are in the same row or the same column
Since triangle {v1, v2, vn} bounds the drawing, this gives: Theorem 3 Every planar graph has a planar straight-line drawing in an n × (2 + (n − 2)(n − 3))-grid such that all vertices have distinct x-coordinates. While this theorem per se is not useful for non-aligned drawings, we find it interesting from a didactic point of view: It proves that polynomial coordinates can be achieved for straight-line drawings of planar graphs, and requires for this only the canonical ordering, but neither the properties of Schnyder trees [20] nor the details of how to “shift” that is needed for other methods using the canonical ordering (e.g. [10, 13])
No planar graph is known that needs more than one bend in a planar rook-drawing, and no planar graph is known that needs more than 2n + 1 grid-lines in a planar non-aligned drawing
Summary
At the GD 2015 conference, Auber et al [3] introduced the concept of rookdrawings: These are drawings of a graph in an n × n-grid such that no two vertices are in the same row or the same column (if the vertices were rooks on a chessboard, no vertex could beat any other) They showed that not all planar graphs have a planar straight-line rook-drawing, and gave a construction of planar rook-drawings with at most n − 3 bends. Modifying the construction a bit, we can achieve that all y-coordinates are distinct and that the height is cubic This is achieved via creating a so-called rectangle-of-influence drawing of a modification of the graph, and arguing that each modification can be undone while adding only one bend.
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