Rook-drawing for plane graphs Claire Pennarun David Auber, Nicolas - PowerPoint PPT Presentation

Rook-drawing for plane graphs Claire Pennarun David Auber, Nicolas Bonichon and Paul Dorbec LaBRI, Bordeaux Graph Drawing September 25 th , 2015 Claire Pennarun (LaBRI, Bordeaux) Rook-drawing for plane graphs GD 2015 1 / 15

Maybe you know them? Claire Pennarun (LaBRI, Bordeaux) Rook-drawing for plane graphs GD 2015 2 / 15

Drawing graphs We want to draw large graphs with hierarchical view: a vertex in the drawing = a group of vertices in the graph Claire Pennarun (LaBRI, Bordeaux) Rook-drawing for plane graphs GD 2015 3 / 15

Drawing graphs We want to draw large graphs with hierarchical view: a vertex in the drawing = a group of vertices in the graph preserve the relative positions of vertices → mental map Claire Pennarun (LaBRI, Bordeaux) Rook-drawing for plane graphs GD 2015 3 / 15

Drawing graphs We want to draw large graphs with hierarchical view: a vertex in the drawing = a group of vertices in the graph preserve the relative positions of vertices → mental map low complexity of algorithms (linear, if possible) Claire Pennarun (LaBRI, Bordeaux) Rook-drawing for plane graphs GD 2015 3 / 15

Drawing graphs We want to draw large graphs with hierarchical view: a vertex in the drawing = a group of vertices in the graph preserve the relative positions of vertices → mental map low complexity of algorithms (linear, if possible) → new type of drawing with constraints: rook-drawing Claire Pennarun (LaBRI, Bordeaux) Rook-drawing for plane graphs GD 2015 3 / 15

R ook-drawing A rook-drawing of a graph of n vertices: Straight-line edges Regular grid ( n columns and rows) Exactly one vertex per row and column n n n n n n n n n n n n n n n n n n n n n n n n Claire Pennarun (LaBRI, Bordeaux) Rook-drawing for plane graphs GD 2015 4 / 15

R ook-drawing A rook-drawing of a graph of n vertices: Straight-line edges Regular grid ( n columns and rows) Exactly one vertex per row and column n n n n n n n n Claire Pennarun (LaBRI, Bordeaux) Rook-drawing for plane graphs GD 2015 4 / 15

R ook-drawing A rook-drawing of a graph of n vertices: Straight-line edges Regular grid ( n columns and rows) Exactly one vertex per row and column n n n n n n n n : n n : : : n n n n n n Claire Pennarun (LaBRI, Bordeaux) Rook-drawing for plane graphs GD 2015 4 / 15

R ook-drawing A rook-drawing of a graph of n vertices: Straight-line edges Regular grid ( n columns and rows) Exactly one vertex per row and column n n n n n n n n : n n : n n : : n n n : : : : n n n n n n n n n Claire Pennarun (LaBRI, Bordeaux) Rook-drawing for plane graphs GD 2015 4 / 15

Planar rook-drawing Is there a planar rook-drawing for every plane graph? Claire Pennarun (LaBRI, Bordeaux) Rook-drawing for plane graphs GD 2015 5 / 15

Planar rook-drawing Is there a planar rook-drawing for every plane graph? A plane graph = a planar graph + an embedding in the plane Claire Pennarun (LaBRI, Bordeaux) Rook-drawing for plane graphs GD 2015 5 / 15

Planar rook-drawing Is there a planar rook-drawing for every plane graph? A plane graph = a planar graph + an embedding in the plane What we already know: Straight-lines drawing ([F´ ary, 1948] : every planar graph) Grid drawing ([de Fraysseix, 1988], [Schnyder, 1990] : every plane graph on an ( n − 2 ) × ( n − 2 ) grid) Claire Pennarun (LaBRI, Bordeaux) Rook-drawing for plane graphs GD 2015 5 / 15

Outer face of degree 3 Three exterior vertices a , b and c , n vertices (here n = 6). b n n c a n Claire Pennarun (LaBRI, Bordeaux) Rook-drawing for plane graphs GD 2015 6 / 15

Outer face of degree 3 Three exterior vertices a , b and c , n vertices (here n = 6). b We must place interior vertices on the red n row and column n c a n Claire Pennarun (LaBRI, Bordeaux) Rook-drawing for plane graphs GD 2015 6 / 15

Outer face of degree 3 Three exterior vertices a , b and c , n vertices (here n = 6). b We must place interior vertices on the red n row and column α α ≥ 45 ◦ , β ≥ 45 ◦ + Mbc right-angled β n c M a n Claire Pennarun (LaBRI, Bordeaux) Rook-drawing for plane graphs GD 2015 6 / 15

Outer face of degree 3 Three exterior vertices a , b and c , n vertices (here n = 6). b We must place interior vertices on the red n row and column α α ≥ 45 ◦ , β ≥ 45 ◦ + Mbc right-angled β n c α = β = 45 ◦ a n Claire Pennarun (LaBRI, Bordeaux) Rook-drawing for plane graphs GD 2015 6 / 15

Outer face of degree 3 Three exterior vertices a , b and c , n vertices (here n = 6). b We must place interior vertices on the red n row and column c n α ≥ 45 ◦ , β ≥ 45 ◦ + Mbc right-angled α = β = 45 ◦ Fill these row and column with c and b ! a n Claire Pennarun (LaBRI, Bordeaux) Rook-drawing for plane graphs GD 2015 6 / 15

Outer face of degree 3 Three exterior vertices a , b and c , n vertices (here n = 6). b We must place interior vertices on the red n row and column c n α ≥ 45 ◦ , β ≥ 45 ◦ + Mbc right-angled n n α = β = 45 ◦ n Fill these row and column with c and b ! a n Inner nodes: along a diagonal Claire Pennarun (LaBRI, Bordeaux) Rook-drawing for plane graphs GD 2015 6 / 15

Outer face of degree 3 Three exterior vertices a , b and c , n vertices (here n = 6). b We must place interior vertices on the red n row and column c n α ≥ 45 ◦ , β ≥ 45 ◦ + Mbc right-angled n n α = β = 45 ◦ n Fill these row and column with c and b ! a n Inner nodes: along a diagonal Graph with a degree 3 outer face with planar rook-drawing = subgraph of the tower graph Claire Pennarun (LaBRI, Bordeaux) Rook-drawing for plane graphs GD 2015 6 / 15

R ook-drawing for outerplanar graphs A graph is outerplanar if it has a planar drawing such that all its vertices are on the outer face. Claire Pennarun (LaBRI, Bordeaux) Rook-drawing for plane graphs GD 2015 7 / 15

R ook-drawing for outerplanar graphs A graph is outerplanar if it has a planar drawing such that all its vertices are on the outer face. n n n n n n n n n n n n n n n n [Bonichon, Gavoille, Hanusse, 2005] edges of G maximal rooted outerplane graph → T b , T r Claire Pennarun (LaBRI, Bordeaux) Rook-drawing for plane graphs GD 2015 7 / 15

R ook-drawing for outerplanar graphs A graph is outerplanar if it has a planar drawing such that all its vertices are on the outer face. n n n n n n n n n n n n n n n n [Bonichon, Gavoille, Hanusse, 2005] edges of G maximal rooted outerplane graph → T b , T r Claire Pennarun (LaBRI, Bordeaux) Rook-drawing for plane graphs GD 2015 7 / 15

R ook-drawing for outerplanar graphs Only look to T b : Claire Pennarun (LaBRI, Bordeaux) Rook-drawing for plane graphs GD 2015 8 / 15

R ook-drawing for outerplanar graphs Only look to T b : 1 x : ccw pre-order depth-first y : ccw post-order depth-first Claire Pennarun (LaBRI, Bordeaux) Rook-drawing for plane graphs GD 2015 8 / 15

R ook-drawing for outerplanar graphs Only look to T b : 1 2 x : ccw pre-order depth-first y : ccw post-order depth-first Claire Pennarun (LaBRI, Bordeaux) Rook-drawing for plane graphs GD 2015 8 / 15

R ook-drawing for outerplanar graphs Only look to T b : 1 2 3 x : ccw pre-order depth-first y : ccw post-order depth-first Claire Pennarun (LaBRI, Bordeaux) Rook-drawing for plane graphs GD 2015 8 / 15

R ook-drawing for outerplanar graphs Only look to T b : 9 13 14 1 16 10 15 12 2 3 11 5 8 4 6 7 x : ccw pre-order depth-first y : ccw post-order depth-first Claire Pennarun (LaBRI, Bordeaux) Rook-drawing for plane graphs GD 2015 8 / 15

R ook-drawing for outerplanar graphs Only look to T b : 9 13 14 1 16 10 15 12 2 1 3 11 5 8 4 6 7 x : ccw pre-order depth-first y : ccw post-order depth-first Claire Pennarun (LaBRI, Bordeaux) Rook-drawing for plane graphs GD 2015 8 / 15

R ook-drawing for outerplanar graphs Only look to T b : 9 13 14 1 16 10 15 12 2 1 3 11 5 2 8 4 6 7 x : ccw pre-order depth-first y : ccw post-order depth-first Claire Pennarun (LaBRI, Bordeaux) Rook-drawing for plane graphs GD 2015 8 / 15

R ook-drawing for outerplanar graphs Only look to T b : 9 13 14 1 16 10 15 12 2 1 3 11 5 2 8 4 6 3 7 x : ccw pre-order depth-first y : ccw post-order depth-first Claire Pennarun (LaBRI, Bordeaux) Rook-drawing for plane graphs GD 2015 8 / 15

R ook-drawing for outerplanar graphs Only look to T b : 9 15 13 14 14 13 1 16 16 12 10 10 15 11 12 9 2 1 3 7 11 8 5 2 8 6 4 5 6 3 7 4 x : ccw pre-order depth-first y : ccw post-order depth-first Claire Pennarun (LaBRI, Bordeaux) Rook-drawing for plane graphs GD 2015 8 / 15

R ook-drawing for outerplanar graphs Only look to T b : n (9,15) n (13,14) n (14,13) n (1,16) (16,12) n n (10,10) (15,11) n (12,9) n (2,1) (3,7) n (11,8) (5,2) n (8,6) n n (6,3) (4,5) n (7,4) n n x : ccw pre-order depth-first n y : ccw post-order depth-first Claire Pennarun (LaBRI, Bordeaux) Rook-drawing for plane graphs GD 2015 8 / 15