1. On Balanced + -Contact Representations Stephane Durocher & Debajyoti Mondal University of Manitoba

2. Contact Graph  Each vertex is represented by a closed region.  The interiors of every pair of vertices are disjoint.  Two vertices are joined by an edge iff the boundaries of their regions touch. Theorem [Koebe 1936] Every planar graph has a circle contact representation. Cover Contact Graph (Circle Contact Representation) a b c d e a b e d c Graph Drawing, Bordeaux. 2 September 23-25, 2013

3. b g Other Shapes Graph Drawing, Bordeaux. 3 September 23-25, 2013 Point-contact of disks (Every Planar Graph) [Koebe 1936] Point-contact of triangles (Every Planar Graph) [ de Fraysseix et al. 1994 ] a b c d e g f f a c d e a b c d e f g Rectangle contact representation (Complete Characterization) [Kozminski & Kinnen 1985, Kant & He 2003] A node-link diagram Side-contact of polygons (octagon  hexagon) [He 1999, Liao et al. 2003, Duncan et al. 2011] a b g c f d e b a d c e f g g point contact side contact

4. +-Contact Representations  Each vertex is represented by an axis-aligned +.  Two + shapes never cross.  Two + shapes touch iff the corresponding vertices are adjacent. Graph Drawing, Bordeaux, France. 4 September 23-25, 2013 a d f g e c b g e f d c b a Allowed Not Allowed

5. c-Balanced + -Contact Representations  Each arm can touch at most ⌈ c∆ ⌉ other arms. Graph Drawing, Bordeaux. 5 September 23-25, 2013 a d f g e c b g e f d c b a A plane graph G with maximum degree ∆ = 5 A (1/2)-balanced +-contact representation of G

6. c-Balanced + -Contact vs. T- and L-Contact  Every planar graph admits a T-contact representation [de Fraysseix et al. 1994]. Several recent attempts to characterize L-contact graphs [Kobourov et al. 2013, Chaplick et al. 2013].  T- and L-contact representations may be unbalanced, but our goal with +- contact is to construct balanced representations. Graph Drawing, Bordeaux. 6 September 23-25, 2013 a b c d f e g a d f g e c b g e f d c b a A plane graph G with ∆ = 5 A (1/2)-balanced +-contact representation A T-contact representation

7. c-Balanced Representations: Applications Graph Drawing, Bordeaux. 7 September 23-25, 2013 a d f g e c b g e f d c b a a d ge f b c g e f d c b a An 1-bend orthogonal drawing with boxes of size ⌈ c∆ ⌉ × ⌈ c∆ ⌉ A transformation into an 1-bend polyline drawing with 2 ⌈ c∆ ⌉ slopes [Keszegh et al, 2000]

8. Results Graph Drawing, Bordeaux. 8 September 23-25, 2013  2-trees have (1/3)-balanced +-contact representations, but not necessarily a (1/4-)-balanced +-contact representation.  Plane 3-trees have (1/2)-balanced +-contact representations, but not necessarily a (1/3)-balanced +-contact representation.  Strengthens the result that 2-trees with ∆ = 4, have 1-bend orthogonal (equivalent to (1/4)-balanced +-contact) [Tayu et al., 2009].  Implies 1-bend polyline drawings of 2-trees with 2 ⌈ ∆/3 ⌉ -slopes, and for plane 3-trees with 2 ⌈ ∆/2 ⌉ slopes, which is significantly smaller than the upper bound of 2∆ for general planar graphs [Keszegh et al. 2010].  It is interesting that with 1-bend per edge, we use roughly 2∆/3 slopes for 2-trees, where the planar slope number of 2-trees is in [∆-3, 2∆] [Lenhart et al., 2013].

9. 2-Trees Graph Drawing, Bordeaux. 9 September 23-25, 2013 A 2-tree (series-parallel graph) G with n ≥ 2 vertices is constructed as follows. Base Case: Series combination : Parallel combination: G1G1 G2G2 G1G1 G2G2 s1s1 t1t1 s2s2 t2t2 s1s1 t1t1 s2s2 t2t2 s1s1 s 1 = s 2 t2t2 t 1 = t 2 t 1 = s 2 poles

10. Series-Parallel Decompositions Graph Drawing, Bordeaux. 10 September 23-25, 2013 S S P P S S P P S S S S P P P P a b c d e f g a b c d e f g

11. Series-Parallel Decompositions Graph Drawing, Bordeaux. 11 September 23-25, 2013 S S P P S S P P S S S S P P P P a b c d e f g

12. (1/2)-Balanced Representation for 2-trees Graph Drawing, Bordeaux. 12 September 23-25, 2013  Let G be a 2-tree, and H=G \ (s,t).  Let f(a) denote the free points of the arm a. Initially, f(a) = ⌈ ∆/2 ⌉ or f(a) = ⌈ ∆/2 ⌉ -1 (if there is an edge (s,t) in G).  Claim: Given a rectangle R=abcd such that degree(s,H) ≤ f(a d ) + f(a r ) and degree(t,H) ≤ f(c l ) + f(c u ), one can construct a (1/2)-balanced representation of H inside R such that s and t lie on a and c, respectively. arar GH R a b c d s s t t adad clcl cucu

13. (1/2)-Balanced Representation for 2-trees Graph Drawing, Bordeaux. 13 September 23-25, 2013  Let G be a 2-tree, and H=G \ (s,t).  Claim: Given a rectangle R=abcd such that degree(s,H) ≤ f(a d ) + f(a r ) and degree(t,H) ≤ f(c l ) + f(c u ), one can construct a (1/2)-balanced representation of H inside R such that s and t lie on a and c, respectively. H R a b c d R a (= s) b c (= t) d Base Case:  H consists of two isolated vertices: s and t. s t

14. (1/2)-Balanced Representation for 2-trees Graph Drawing, Bordeaux. 14 September 23-25, 2013  Let G be a 2-tree, and H=G \ (s,t).  Claim: Given a rectangle R=abcd such that degree(s,H) ≤ f(a d ) + f(a r ) and degree(t,H) ≤ f(c l ) + f(c u ), one can construct a (1/2)-balanced representation of H inside R such that s and t lie on a and c, respectively. H R a b c d Series Combination  Induction: Draw H 1 \ (s 1,t 1 ) and H 2 \ (s 2,t 2 ) inside R 1 and R 2, respectively. s = s 1 t = t 2 t 1 = s 2 H1H1 H2H2 a (= s) b c (= t) d m ( = t 1 ) R1R1 R2R2 f(a d )-1 ⌈ ∆/2 ⌉ -1 a1a1 ⌈ ∆/2 ⌉ m1m1 f(ar)f(ar) m2m2 f(cl)f(cl) f(c u )-1 c2c2

15. (1/2)-Balanced Representation for 2-trees Graph Drawing, Bordeaux. 15 September 23-25, 2013 a (= s) b c (= t) d m ( = t 1 ) R1R1 R2R2 f(a d )-1 ⌈ ∆/2 ⌉ -1 a1a1 ⌈ ∆/2 ⌉ m1m1 f(ar)f(ar) m2m2 f(c l )-1 f(cu)f(cu) c2c2 a (= s) b c (= t) d m ( = t 1 ) R1R1 R2R2 f(a d )-1 ⌈ ∆/2 ⌉ -1 a1a1 ⌈ ∆/2 ⌉ ⌈ ∆/2 ⌉ -1 m1m1 f(ar)f(ar) m2m2 f(cl)f(cl) f(c u )-1 c2c2 Series Combination  Induction: Draw H 1 \ (s 1,t 1 ) and H 2 \ (s 2,t 2 ) inside R 1 and R 2, respectively.

16. (1/2)-Balanced Representation for 2-trees Graph Drawing, Bordeaux. 16 September 23-25, 2013  Let G be a 2-tree, and H=G \ (s,t).  Claim: Given a rectangle R=abcd such that degree(s,H) ≤ f(a d ) + f(a r ) and degree(t,H) ≤ f(c l ) + f(c u ), one can construct a (1/2)-balanced representation of H inside R such that s and t lie on a and c, respectively. H R a b c d a1a1 b c1c1 d Parallel Combination  Distribute the free points of R among R 1 and R 2. H1H1 H2H2 s 1 = s 2 t 1 = t 2 R1R1 a2a2 b c (= t 1 ) d R2R2

17. (1/2)-Balanced Representation for 2-trees Graph Drawing, Bordeaux. 17 September 23-25, 2013  Let G be a 2-tree, and H=G \ (s,t).  Claim: Given a rectangle R=abcd such that degree(s,H) ≤ f(a d ) + f(a r ) and degree(t,H) ≤ f(c l ) + f(c u ), one can construct a (1/2)-balanced representation of H inside R such that s and t lie on a and c, respectively. H R a b c d 10 b1b1 c1c1 d1d1 H1H1 H2H2 s 1 = s 2 t 1 = t 2 R1R1 a2a2 b2b2 c2c2 d2d2 R2R2 5 0 21 10 5 0 3 0 16 0 2 degree (s 1,H 1 ) = 15 degree (t 1,H 1 ) = 3 a1a1 Parallel Combination  Distribute the free points of R among R 1 and R 2.

18. (1/2)-Balanced Representation for 2-trees Graph Drawing, Bordeaux. 18 September 23-25, 2013  Let G be a 2-tree, and H=G \ (s,t).  Claim: Given a rectangle R=abcd such that degree(s,H) ≤ f(a d ) + f(a r ) and degree(t,H) ≤ f(c l ) + f(c u ), one can construct a (1/2)-balanced representation of H inside R such that s and t lie on a and c, respectively. H R a b c d Parallel Combination  Draw H 1 and H 2 using induction, and merge them avoiding edge crossing. H1H1 H2H2 s 1 = s 2 t 1 = t 2

19. (1/2)-Balanced Representation for 2-trees Graph Drawing, Bordeaux. 19 September 23-25, 2013  Let G be a 2-tree, and H=G \ (s,t).  Claim: Given a rectangle R=abcd such that degree(s,H) ≤ f(a d ) + f(a r ) and degree(t,H) ≤ f(c l ) + f(c u ), one can construct a (1/2)-balanced representation of H inside R such that s and t lie on a and c, respectively. We started with G and proved that H admits a (1/2)-balanced +-contact representation inside R. If the poles of G are adjacent, we initialize f( a d )= ⌈ ∆/2 ⌉ -1 and f( c l )= ⌈ ∆/2 ⌉ -1, then draw H. Finally, draw (s,t) along abc. arar GH R a b c d s s t t adad clcl cucu G a b c d H a b c d

20. Refinement: (1/3)-Balanced Representation Graph Drawing, Bordeaux. 20 September 23-25, 2013  Why was the previous construction (1/2)-balanced?  While adding a new arm, we assigned at most ⌈ ∆/2 ⌉ free points to it.  Since ⌈ ∆/2 ⌉ + ⌈ ∆/2 ⌉ ≥ ∆, we could find a ‘nice’ rectangle partition, i.e., using at most two arms.  Recall series combination. a (= s) b c (= t) d m R1R1 R2R2 ⌈ ∆/2 ⌉ -1 a1a1 ⌈ ∆/2 ⌉ m1m1 m2m2 c2c2

21.  For (1/3)-balanced we assign at most ⌈ ∆/3 ⌉ free points to any arm.  Sometimes we need at least three of the arms of m to lie in the same rectangle. E.g., if degree(m,H 1 ) > 2 ⌈ ∆/3 ⌉.  Sometimes we need to share an arm among the rectangles. E.g., assume degree(m,H 1 ) > ⌈ ∆/3 ⌉ and degree(m,H 2 ) > ⌈ ∆/3 ⌉ in the following. Refinement: (1/3)-Balanced Representation Graph Drawing, Bordeaux. 21 September 23-25, 2013 a (= s) b c (= t) d m ⌈ ∆/3 ⌉ -1 a1a1 ⌈ ∆/3 ⌉ 0 m1m1 m2m2 c2c2 a (= s) b c (= t) d m ⌈ ∆/3 ⌉ -1 a1a1 m1m1 m2m2 c2c2 deg ( m 1, H 1 ) – ( ⌈ ∆/3 ⌉ -1) deg ( m 2,H 2 ) – ( ⌈ ∆/3 ⌉ -1) 0

22. Refinement: (1/3)-Balanced Representation Graph Drawing, Bordeaux. 22 September 23-25, 2013 a (= s) b c (= t) d m ⌈ ∆/3 ⌉ -1 a1a1 ⌈ ∆/3 ⌉ m1m1 m2m2 c2c2 a (= s) b c (= t) d m ⌈ ∆/3 ⌉ -1 a1a1 m1m1 m2m2 c2c2 deg ( m 1 ) – ( ⌈ ∆/3 ⌉ -1) outnDeg ( m 2 ) – ( ⌈ ∆/3 ⌉ -1) 0 Some poles do not lie at the corners. More case Analysis! Some poles do not lie at the corners. More case Analysis!

23. (1/3)-Balanced Representation for 2-trees Graph Drawing, Bordeaux. 23 September 23-25, 2013 a (= s) b c (= t) d m ⌈ ∆/3 ⌉ -1 a1a1 ⌈ ∆/3 ⌉ m1m1 m2m2 c2c2 a (= s) b c (= t) d m ⌈ ∆/3 ⌉ -1 m2m2 c2c2 outnDeg ( m 2 ) – ( ⌈ ∆/3 ⌉ -1) 0 ⌈ ∆/3 ⌉ -1 a1a1 m1m1 deg ( m 1 ) – ( ⌈ ∆/3 ⌉ -1) Sometimes flip sub- problems to apply induction. a1a1 m1m1 Some poles do not lie at the corners. More case Analysis! Some poles do not lie at the corners. More case Analysis!

24. Plane 3-trees: (1/2)-Balanced Graph Drawing, Bordeaux. 24 September 23-25, 2013 a b c p a b p b c p a c p

25. Conclusion Graph Drawing, Bordeaux. 25 September 23-25, 2013  Summary  2-trees have (1/3)-balanced +-contact representations, but not necessarily a (1/4- )-balanced +-contact representation.  Plane 3-trees have (1/2)-balanced +-contact representations, but not necessarily a (1/3)-balanced +-contact representation.  Open Questions  Although our representations for planar 3-trees preserve the input embedding, our representations for 2-trees do not have this property. Do there exist algorithms for (1/3)-balanced representations of 2-trees that preserve input embedding?  Close the gap between the lower and upper bounds.  Characterize planar graphs that admit c-balanced +-contact representations, for small fixed values c.

26. Thank You September 23-25, 2013 26 Graph Drawing, Bordeaux.