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On Balanced + -Contact Representations

On Balanced + -Contact Representations. Stephane Durocher & Debajyoti Mondal University of Manitoba. Contact Graph. a. c. a. c. e. d. e. d. b. b. Cover Contact Graph (Circle Contact Representation). Theorem [ Koebe 1936] Every planar graph has a circle contact representation.

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On Balanced + -Contact Representations

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

  2. Contact Graph a c a c e d e d b b Cover Contact Graph (Circle Contact Representation) Theorem [Koebe 1936] Every planar graph has a circle contact representation. 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. Graph Drawing, Bordeaux.

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

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

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

  6. c-Balanced +-Contact vs. T- and L-Contact f f g a d d g e e c c a b b f A (1/2)-balanced +-contact representation A T-contact representation A plane graph G with ∆ = 5 g e d c b a Graph Drawing, Bordeaux. 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.

  7. c-Balanced Representations: Applications f f g a d d g e e c c a b b f a d g e f a c d e g b c 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] b Graph Drawing, Bordeaux.

  8. Results Graph Drawing, Bordeaux. • 2-trees have (1/3)-balanced +-contact representations, but not necessarily a (1/4-ϵ)-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) [Tayuet al., 2009]. • Implies 1-bend polyline drawings of 2-trees with 2⌈∆/3⌉-slopes, and for plane3-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 A 2-tree (series-parallel graph) G with n ≥ 2vertices is constructed as follows. Base Case: s1 s1 t1 t1 t2 t1= s2 s1 G1 Series combination : s2 s2 t2 t2 G2 poles s1= s2 t1= t2 G1 Parallel combination: G2 poles Graph Drawing, Bordeaux.

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

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

  12. (1/2)-Balanced Representation for 2-trees s s ar d a R ad cu t t c b G H cl 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(ad) + f(ar) and degree(t,H) ≤ f(cl) + f(cu), one can construct a (1/2)-balanced representation of H inside R such that s and t lie on a and c, respectively. Graph Drawing, Bordeaux.

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

  14. (1/2)-Balanced Representation for 2-trees f(ar) a1 a (= s) t1= s2 t = t2 s = s1 d R1 ⌈∆/2⌉ m ( = t1) f(ad)-1 H1 m2 ⌈∆/2⌉ m1 d a f(cu)-1 R2 ⌈∆/2⌉-1 R ⌈∆/2⌉ -1 c (=t) b c2 f(cl) H2 c b • Series Combination • Induction: DrawH1\ (s1,t1)and H2 \ (s2,t2)inside R1 and R2, respectively. H Let G be a 2-tree, and H=G \ (s,t). Claim: Given a rectangle R=abcd such that degree(s,H) ≤ f(ad) + f(ar) and degree(t,H) ≤ f(cl) + f(cu), one can construct a (1/2)-balanced representation of H inside R such that s and t lie on a and c, respectively. Graph Drawing, Bordeaux.

  15. (1/2)-Balanced Representation for 2-trees f(ar) f(ar) a1 a1 a (= s) a (= s) d d R1 R1 ⌈∆/2⌉ ⌈∆/2⌉ m ( = t1) m ( = t1) f(ad)-1 f(ad)-1 m2 m2 ⌈∆/2⌉-1 ⌈∆/2⌉ m1 m1 f(cu)-1 f(cu) R2 R2 ⌈∆/2⌉-1 ⌈∆/2⌉ ⌈∆/2⌉ -1 ⌈∆/2⌉ -1 c (=t) c (=t) b b c2 c2 f(cl)-1 f(cl) • Series Combination • Induction: DrawH1\ (s1,t1)and H2 \ (s2,t2)inside R1 and R2, respectively. Graph Drawing, Bordeaux.

  16. (1/2)-Balanced Representation for 2-trees a2 a1 d d H1 R2 R1 s1= s2 t1= t2 d a R c (=t1) c1 b b H2 c b • Parallel Combination • Distribute the free points of R among • R1 andR2. H Let G be a 2-tree, and H=G \ (s,t). Claim: Given a rectangle R=abcd such that degree(s,H) ≤ f(ad) + f(ar) and degree(t,H) ≤ f(cl) + f(cu), one can construct a (1/2)-balanced representation of H inside R such that s and t lie on a and c, respectively. Graph Drawing, Bordeaux.

  17. (1/2)-Balanced Representation for 2-trees a2 a1 5 degree (s1,H1) = 15 16 d1 d2 degree (t1,H1) = 3 21 H1 R1 R2 3 10 s1= s2 t1= t2 2 0 d a R 10 5 c1 c2 b1 b2 0 0 H2 0 c b • Parallel Combination • Distribute the free points of R among • R1 andR2. H Graph Drawing, Bordeaux. Let G be a 2-tree, and H=G \ (s,t). Claim: Given a rectangle R=abcd such that degree(s,H) ≤ f(ad) + f(ar) and degree(t,H) ≤ f(cl) + f(cu), one can construct a (1/2)-balanced representation of H inside R such that s and t lie on a and c, respectively.

  18. (1/2)-Balanced Representation for 2-trees H1 s1= s2 t1= t2 d a R H2 c b H • Parallel Combination • Draw H1 and H2 using induction, and merge them avoiding edge crossing. Graph Drawing, Bordeaux. Let G be a 2-tree, and H=G \ (s,t). Claim: Given a rectangle R=abcd such that degree(s,H) ≤ f(ad) + f(ar) and degree(t,H) ≤ f(cl) + f(cu), one can construct a (1/2)-balanced representation of H inside R such that s and t lie on a and c, respectively.

  19. (1/2)-Balanced Representation for 2-trees ar s s d d a a d a H G R ad cu c c b b c t t b G H cl 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(ad)=⌈∆/2⌉ -1 and f(cl)=⌈∆/2⌉ -1, then draw H. Finally,draw (s,t) along abc. Graph Drawing, Bordeaux. Let G be a 2-tree, and H=G \ (s,t). Claim: Given a rectangle R=abcd such that degree(s,H) ≤ f(ad) + f(ar) and degree(t,H) ≤ f(cl) + f(cu), one can construct a (1/2)-balanced representation of H inside R such that s and t lie on a and c, respectively.

  20. Refinement: (1/3)-Balanced Representation a1 a (= s) d R1 ⌈∆/2⌉ m2 m ⌈∆/2⌉ m1 R2 ⌈∆/2⌉-1 ⌈∆/2⌉ -1 c (=t) b c2 • 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. Graph Drawing, Bordeaux.

  21. Refinement: (1/3)-Balanced Representation a1 a (= s) a1 a (= s) d d m2 m ⌈∆/3⌉ deg(m2,H2) – (⌈∆/3⌉-1) m2 ⌈∆/3⌉ -1 0 m m1 m1 ⌈∆/3⌉ ⌈∆/3⌉ -1 ⌈∆/3⌉-1 deg(m1,H1) – (⌈∆/3⌉-1) ⌈∆/3⌉-1 c (=t) b c2 c (=t) b c2 0 Graph Drawing, Bordeaux. • 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,H1) > 2⌈∆/3⌉ . • Sometimes we need to share an arm among the rectangles. E.g., assume degree(m,H1) > ⌈∆/3⌉ and degree(m,H2) > ⌈∆/3⌉ in the following.

  22. Refinement: (1/3)-Balanced Representation Some poles do not lie at the corners. More case Analysis! a1 a (= s) a1 a (= s) d d m2 m ⌈∆/3⌉ outnDeg(m2) – (⌈∆/3⌉-1) m2 ⌈∆/3⌉ -1 ⌈∆/3⌉ m m1 m1 ⌈∆/3⌉ ⌈∆/3⌉ -1 ⌈∆/3⌉-1 deg(m1) – (⌈∆/3⌉-1) ⌈∆/3⌉-1 c (=t) b c2 c (=t) b c2 0 Graph Drawing, Bordeaux.

  23. (1/3)-Balanced Representation for 2-trees Some poles do not lie at the corners. More case Analysis! a1 a (= s) d m2 m ⌈∆/3⌉ ⌈∆/3⌉ m1 ⌈∆/3⌉ ⌈∆/3⌉ -1 ⌈∆/3⌉-1 c (=t) b c2 Sometimes flip sub-problems to apply induction. a (= s) d outnDeg(m2) – (⌈∆/3⌉-1) m2 m 0 ⌈∆/3⌉-1 a1 c (=t) b a1 c2 ⌈∆/3⌉ -1 m1 m1 deg(m1) – (⌈∆/3⌉-1) Graph Drawing, Bordeaux.

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

  25. Conclusion Graph Drawing, Bordeaux. • Summary • 2-trees have (1/3)-balanced +-contact representations, but not necessarily a (1/4-ϵ)-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, ourrepresentations 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 Graph Drawing, Bordeaux.

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