Drawing graphs with vertices and edges in convex position and large - PowerPoint PPT Presentation

Drawing graphs with vertices and edges in convex position and large polygons in Minkowski sums Kolja Knauer Ignacio Garc´ ıa-Marco LIP ENS Lyon LIF Marseille Graph Drawing , September 29, 2015

P ⊆ R 2 in strictly convex position if P is the set of vertices of the convex hull of P

P ⊆ R 2 in strictly convex position if P is the set of vertices of the convex hull of P

P ⊆ R 2 in strictly convex position if P is the set of vertices of the convex hull of P

P ⊆ R 2 in strictly convex position if P is the set of vertices of the convex hull of P

P ⊆ R 2 in strictly convex P ⊆ R 2 in weakly convex position if P is the set of vertices position if P on the ”boundary” of the convex hull of P of the convex hull of P

P ⊆ R 2 in strictly convex P ⊆ R 2 in weakly convex position if P is the set of vertices position if P on the ”boundary” of the convex hull of P of the convex hull of P → R 2 such that: consider graph drawings f : G ֒ ◦ edges straight-line segments ◦ vertices and midpoints of edges on different points

P ⊆ R 2 in strictly convex P ⊆ R 2 in weakly convex position if P is the set of vertices position if P on the ”boundary” of the convex hull of P of the convex hull of P ok → R 2 such that: consider graph drawings f : G ֒ ◦ edges straight-line segments ◦ vertices and midpoints of edges on different points

P ⊆ R 2 in strictly convex P ⊆ R 2 in weakly convex position if P is the set of vertices position if P on the ”boundary” of the convex hull of P of the convex hull of P ok → R 2 such that: consider graph drawings f : G ֒ ◦ edges straight-line segments ◦ vertices and midpoints of edges on different not ok points

P ⊆ R 2 in strictly convex P ⊆ R 2 in weakly convex position if P is the set of vertices position if P on the ”boundary” of the convex hull of P of the convex hull of P ok → R 2 such that: consider graph drawings f : G ֒ ◦ edges straight-line segments ◦ vertices and midpoints of edges on different not ok points   strictly convex if j = s  midpoints position weakly convex if j = w   arbitrary if j = a. for i, j ∈ { s, w, a } define G j  i as class of graphs drawable s.th.  strictly convex if i = s  vertex position weakly convex if i = w   arbitrary if i = a.

Theorem [G-M,K] : G w w = G w a = G a s = G a w = G a a G s G w a s G s s = G s w   strictly convex if j = s  midpoints position weakly convex if j = w   arbitrary if j = a. for i, j ∈ { s, w, a } define G j  i as class of graphs drawable s.th.  strictly convex if i = s  vertex position weakly convex if i = w   arbitrary if i = a.

Theorem [G-M,K] : G w w = G w a = G a s = G a w = G a a G s G w a s G s s = G s w   strictly convex if j = s  midpoints position weakly convex if j = w   arbitrary if j = a. for i, j ∈ { s, w, a } define G j  i as class of graphs drawable s.th.  strictly convex if i = s  vertex position weakly convex if i = w   arbitrary if i = a.

g j i ( n ) max number of edges n -vertex graph in G j i Conjecture [Halmann, Onn, Rothblum 07] : All graphs in G s s are planar and therefore g s s ( n ) ≤ 3 n − 6 .

g j i ( n ) max number of edges n -vertex graph in G j i Conjecture [Halmann, Onn, Rothblum 07] : All graphs in G s s are planar and therefore g s s ( n ) ≤ 3 n − 6 . , . . . ∈ G s , , , s

g j i ( n ) max number of edges n -vertex graph in G j i Conjecture [Halmann, Onn, Rothblum 07] : All graphs in G s s are planar and therefore g s s ( n ) ≤ 3 n − 6 . , . . . ∈ G s , , , s non-planar

g j i ( n ) max number of edges n -vertex graph in G j i Conjecture [Halmann, Onn, Rothblum 07] : All graphs in G s s are planar and therefore g s s ( n ) ≤ 3 n − 6 . , . . . ∈ G s , , , s , , ,

g j i ( n ) max number of edges n -vertex graph in G j i Conjecture [Halmann, Onn, Rothblum 07] : All graphs in G s s are planar and therefore g s s ( n ) ≤ 3 n − 6 . , . . . ∈ G s , , , s , , ,

g j i ( n ) max number of edges n -vertex graph in G j i Conjecture [Halmann, Onn, Rothblum 07] : All graphs in G s s are planar and therefore g s s ( n ) ≤ 3 n − 6 . , . . . ∈ G s , , , s , , ,

g j i ( n ) max number of edges n -vertex graph in G j i Conjecture [Halmann, Onn, Rothblum 07] : All graphs in G s s are planar and therefore g s s ( n ) ≤ 3 n − 6 . , . . . ∈ G s , , , s , , ,

g j i ( n ) max number of edges n -vertex graph in G j i Conjecture [Halmann, Onn, Rothblum 07] : All graphs in G s s are planar and therefore g s s ( n ) ≤ 3 n − 6 . , . . . ∈ G s , , , s , , ,

g j i ( n ) max number of edges n -vertex graph in G j i Conjecture [Halmann, Onn, Rothblum 07] : All graphs in G s s are planar and therefore g s s ( n ) ≤ 3 n − 6 . , . . . ∈ G s , , , s , , ,

g j i ( n ) max number of edges n -vertex graph in G j i Conjecture [Halmann, Onn, Rothblum 07] : All graphs in G s s are planar and therefore g s s ( n ) ≤ 3 n − 6 . , . . . ∈ G s , , , s , , ,

g j i ( n ) max number of edges n -vertex graph in G j i Conjecture [Halmann, Onn, Rothblum 07] : All graphs in G s s are planar and therefore g s s ( n ) ≤ 3 n − 6 . , . . . ∈ G s , , , s , , ,

g j i ( n ) max number of edges n -vertex graph in G j i Conjecture [Halmann, Onn, Rothblum 07] : All graphs in G s s are planar and therefore g s s ( n ) ≤ 3 n − 6 . , . . . ∈ G s , , , s , , ,

g j i ( n ) max number of edges n -vertex graph in G j i Conjecture [Halmann, Onn, Rothblum 07] : All graphs in G s s are planar and therefore g s s ( n ) ≤ 3 n − 6 . , . . . ∈ G s , , , s , , , so, this works...and I wont finish these drawings...

g j i ( n ) max number of edges n -vertex graph in G j i Conjecture [Halmann, Onn, Rothblum 07] : All graphs in G s s are planar and therefore g s s ( n ) ≤ 3 n − 6 .

g j i ( n ) max number of edges n -vertex graph in G j i Conjecture [Halmann, Onn, Rothblum 07] : All graphs in G s s are planar and therefore g s s ( n ) ≤ 3 n − 6 . Theorem [G-M,K] : We have ⌊ 3 2 n − 1 ⌋ ≤ g s s ( n ) ≤ g w s ( n ) = 2 n − 3 .

g j i ( n ) max number of edges n -vertex graph in G j i Conjecture [Halmann, Onn, Rothblum 07] : All graphs in G s s are planar and therefore g s s ( n ) ≤ 3 n − 6 . Theorem [G-M,K] : We have ⌊ 3 2 n − 1 ⌋ ≤ g s s ( n ) ≤ g w s ( n ) = 2 n − 3 . g w s ( n ) ≤ 2 n − 3 :

g j i ( n ) max number of edges n -vertex graph in G j i Conjecture [Halmann, Onn, Rothblum 07] : All graphs in G s s are planar and therefore g s s ( n ) ≤ 3 n − 6 . Theorem [G-M,K] : We have ⌊ 3 2 n − 1 ⌋ ≤ g s s ( n ) ≤ g w s ( n ) = 2 n − 3 . g w s ( n ) ≤ 2 n − 3 : v sees vw and w does not v w

g j i ( n ) max number of edges n -vertex graph in G j i Conjecture [Halmann, Onn, Rothblum 07] : All graphs in G s s are planar and therefore g s s ( n ) ≤ 3 n − 6 . Theorem [G-M,K] : We have ⌊ 3 2 n − 1 ⌋ ≤ g s s ( n ) ≤ g w s ( n ) = 2 n − 3 . g w s ( n ) ≤ 2 n − 3 : v sees vw and w does not v w v v has at most 2 exterior edges

g j i ( n ) max number of edges n -vertex graph in G j i Conjecture [Halmann, Onn, Rothblum 07] : All graphs in G s s are planar and therefore g s s ( n ) ≤ 3 n − 6 . Theorem [G-M,K] : We have ⌊ 3 2 n − 1 ⌋ ≤ g s s ( n ) ≤ g w s ( n ) = 2 n − 3 . g w s ( n ) ≤ 2 n − 3 : v sees vw and w does not v w v v has at most 2 exterior edges v

g j i ( n ) max number of edges n -vertex graph in G j i Conjecture [Halmann, Onn, Rothblum 07] : All graphs in G s s are planar and therefore g s s ( n ) ≤ 3 n − 6 . Theorem [G-M,K] : We have ⌊ 3 2 n − 1 ⌋ ≤ g s s ( n ) ≤ g w s ( n ) = 2 n − 3 . g w s ( n ) ≤ 2 n − 3 : v sees vw and w does not v w v v has at most 2 exterior edges v

g j i ( n ) max number of edges n -vertex graph in G j i Conjecture [Halmann, Onn, Rothblum 07] : All graphs in G s s are planar and therefore g s s ( n ) ≤ 3 n − 6 . Theorem [G-M,K] : We have ⌊ 3 2 n − 1 ⌋ ≤ g s s ( n ) ≤ g w s ( n ) = 2 n − 3 . g w s ( n ) ≤ 2 n − 3 : v sees vw and w does not v w v v has at most 2 exterior edges v v doesn’t see its interior edges

g j i ( n ) max number of edges n -vertex graph in G j i Conjecture [Halmann, Onn, Rothblum 07] : All graphs in G s s are planar and therefore g s s ( n ) ≤ 3 n − 6 . Theorem [G-M,K] : We have ⌊ 3 2 n − 1 ⌋ ≤ g s s ( n ) ≤ g w s ( n ) = 2 n − 3 . g w s ( n ) ≤ 2 n − 3 : v sees vw and w does not v w v v has at most 2 exterior edges v v doesn’t see its interior edges every edge is seen at least once v w

g j i ( n ) max number of edges n -vertex graph in G j i Conjecture [Halmann, Onn, Rothblum 07] : All graphs in G s s are planar and therefore g s s ( n ) ≤ 3 n − 6 . Theorem [G-M,K] : We have ⌊ 3 2 n − 1 ⌋ ≤ g s s ( n ) ≤ g w s ( n ) = 2 n − 3 . g w s ( n ) ≤ 2 n − 3 : v sees vw and w does not v w v v has at most 2 exterior edges v v doesn’t see its interior edges every edge is seen at least once v w = ⇒ every edge is exterior at least once