Victoria city and Sendai city 7300 km Sendai

Victoria city and Sendai city 7300 km Sendai city Victoria city 1

2 Tohoku University was established in 1907. Spring Summer Autumu Winter

GSIS, Tohoku University 3 Graduate School of Information Sciences (GSIS), Tohoku University, was established in 1993. 150 Faculties 450 students Math. Computer Science Robotics Transportation Economics Human Social Sciences Interdisciplinary School

Book 4

5 Small Grid Drawings of Planar Graphs with Balanced Bipartition Xiao Zhou Takashi Hikino Takao Nishizeki Graduate School of Information Sciences, Tohoku University, Japan

6 Grid drawing In a grid drawing of a planar graph, ・ every vertex is located at a grid point, ・ every edge is drawn as a straight-line segment without any edge-intersection. Planar graph Grid drawing 2 3 4 5 7 4 1 1 5 6 7 6

7 Embedding We deal with grid drawings of a planar graph in variable embedding setting. 4 6 Grid drawing 2 2 3 4 7 3 7 Planar graph 3 2 5 This embedding is different from a given embedding 5 1 4 1 1 5 6 7 6

Width and Height of grid drawing H H W W W=9, H=11 W=4, H=3 Area W×H=99 Area W×H=12 8

9 Small grid drawing Large area Small area H H W W We wish to find a small grid drawing in variable embedding setting.

10 Known results Grid drawing of Plane graph Width and Height Running time [Schnyder, 1990] [Chrobak, Kant, 1997] W=n－2, H=n－2 O(n) n ： number of vertices

11 Our results Grid drawing of Plane graph Width and Height Running time [Schnyder, 1990] [Chrobak, Kant, 1997] W=n－2, H=n－2 O(n) If a planar graph G has a balanced bipartition, then G has a grid drawing with small area. s G 2 G 1 Bipartition t Planar graph G s G 1 G 2 t Subgraph G 1, G 2 s t Drawing u 1 G 2 u 2 t Drawing of G

12 Outline of algorithm s G 2 G 1 s Bipartition t G 2 t G 1 (1) Planar graph G Maximal planar graph t s (2) Subgraph G 1, G 2 u 1 G 1 s G 2 u 2 t t (3) Maximal planar graph G 1, G 2 s u 1 G 2 u 2 s Combining t (5) Drawing of G u 1 s G 2 G 1 t u 2 t (4) Drawing of G 1, G 2 g D in w a r

13 Our results s G G 2 G 1 Theorem 1 t (1) Planar graph G If a planar graph G has a balanced bipartition, then G has a grid drawing with small area. W≤max{n 1, n 2}－1 s u 1 G 2 u 2 H≤max{n 1, n 2}－2 t (5) Drawing of G G 1, G 2 Width, Height n 1, n 2≤n/2 W, H≤n/2 n 1, n 2≤ 2 n/3 W, H≤ 2 n/3 n 1, n 2≤αn W, H≤αn If α<1, Balanced bipartition

14 Our results Theorem 1 If a planar graph G has a balanced bipartition, then G has a grid drawing with small area. This graph has NO balanced bipartition. G 1, G 2 Width, Height n 1, n 2≤n/2 W, H≤n/2 n 1, n 2≤ 2 n/3 W, H≤ 2 n/3 n 1, n 2≤αn W, H≤αn If α<1, Balanced bipartition

15 Our results Theorem 1 If a planar graph G has a balanced bipartition, then G has a grid drawing with small area. This graph has NO balanced bipartition. Planar graph Series-Parallel graph Lemma 1 Every Series-Parallel graph has a balanced bipartition (n 1, n 2≤ 2 n/3). α=2/3

16 Our results Theorem 1 If a planar graph G has a balanced bipartition, then G has a grid drawing with small area. This graph has NO balanced bipartition. Lemma 1 Every Series-Parallel graph has a balanced bipartition (n 1, n 2≤ 2 n/3). Planar graph Series-Parallel graph α=2/3 Theorem 2 Series-Parallel W graph H

17 Outline of algorithm s G 2 G 1 s Bipartition t G 2 t G 1 (1) Planar graph G Maximal planar graph t s (2) Subgraph G 1, G 2 u 1 G 1 s G 2 u 2 t t (3) Maximal planar graph G 1, G 2 s u 1 G 2 u 2 s Combining t (5) Drawing of G u 1 s G 2 G 1 t u 2 t (4) Drawing of G 1, G 2 g D in w a r

18 Bipartition We call a pair of distinct vertices {s, t} in a graph G=(V, E) a separation pair of G if G has two subgraphs G 1=(V 1, E 1) and G 2=(V 2, E 2) such that ・ V=V 1∪V 2，V 1∩V 2={s, t}, ・ E=E 1∪E 2，E 1∩E 2=∅． Such a pair of subgraphs {G 1, G 2} is called a bipartition of G. s s G 2 s n 1=9 Bipartition t t G 1 n=12 (1) Graph G (2) Subgraph G 1, G 2 t n 2=5

19 Bipartition We call a pair of distinct vertices {s, t} in a graph G=(V, E) a separation pair of G if G has two subgraphs G 1=(V 1, E 1) and G 2=(V 2, E 2) such that ・ V=V 1∪V 2，V 1∩V 2={s, t}, ・ E=E 1∪E 2，E 1∩E 2=∅． Such a pair of subgraphs {G 1, G 2} is called a bipartition of G. t t t G 2 G 1 Bipartition s n=12 s s n 1=3 (1) Graph G n 2=11 (2) Subgraph G 1, G 2

20 Outline of algorithm s G 2 G 1 s Bipartition t G 2 t G 1 (1) Planar graph G Maximal planar graph t s (2) Subgraph G 1, G 2 u 1 G 1 s G 2 u 2 t t (3) Maximal planar graph G 1, G 2 s u 1 G 2 u 2 s Combining t (5) Drawing of G u 1 s G 2 G 1 t u 2 t (4) Drawing of G 1, G 2 g D in w a r

21 Theorem 1. Let G be a planar graph, and let {G 1, G 2} be an arbitrary bipartition of G. Then G has a grid drawing such that W≤max{n 1, n 2}－1, H≤max{n 1, n 2}－2 and such a drawing can be found in linear time. W≤max{n 1, n 2}－1 s G 2 G 1 t Planar graph G Drawing in linear time u 1 G 2 u 2 t Drawing of G H≤max{n 1, n 2}－2

22 Theorem 1. Let G be a planar graph, and let {G 1, G 2} be an arbitrary bipartition of G. Then G has a grid drawing such that W≤max{n 1, n 2}－1, H≤max{n 1, n 2}－2 and such a drawing can be found in linear time. Assume w. l. o. g. that n 1≥n 2． s s s G 1 t G n=12 t t G 1 n 1=9 n 2=5 G 2

23 Theorem 1. Let G be a planar graph, and let {G 1, G 2} be an arbitrary bipartition of G. Then G has a grid drawing such that W≤max{n 1, n 2}－1, H≤max{n 1, n 2}－2 and such a drawing can be found in linear time. Assume w. l. o. g. that n 1≥n 2． Add dummy edges to G 1 so that the resulting graph is maximal planar and has an edge (s, t). s s G 1 t G n=12 t G 1 n 1=9

24 Theorem 1. Let G be a planar graph, and let {G 1, G 2} be an arbitrary bipartition of G. Then G has a grid drawing such that W≤max{n 1, n 2}－1, H≤max{n 1, n 2}－2 and such a drawing can be found in linear time. Embed G 1 so that the edge (s, t) lies on the outer face of G 1. s s G 1 t G n=12 u 1 G 1 t n 1=9

25 Theorem 1. Let G be a planar graph, and let {G 1, G 2} be an arbitrary bipartition of G. Then G has a grid drawing such that W≤max{n 1, n 2}－1, H≤max{n 1, n 2}－2 and such a drawing can be found in linear time. Obtain a grid drawing of G 1[CK 97]. s s G 1 t G n=12 u 1 G 1 t n 1=9

26 Theorem 1. Let G be a planar graph, and let {G 1, G 2} be an arbitrary s bipartition of G. Then G has a grid drawing such that W≤max{n 1, n 2}－1, H≤max{n 1, n 2}－2 and such a drawing can be found in linear time. Obtain a grid drawing of G 1[CK 97]. u 1 G 1 s H 1=n 1－2 Edge (u 1, t) is horizontal. G 1 n 1=9 u 1 t W 1=n 1－2 t n 1=9

27 Theorem 1. Let G be a planar graph, and let {G 1, G 2} be an arbitrary bipartition of G. Then G has a grid drawing such that W≤max{n 1, n 2}－1, H≤max{n 1, n 2}－2 and such a drawing can be found in linear time. u 2 s s G 2 t G n=12 t u 1 t G 1 n 1=9 n 2=5 G 2

28 Theorem 1. Let G be a planar graph, and let {G 1, G 2} be an arbitrary bipartition of G. Then G has a grid drawing such that W≤max{n 1, n 2}－1, H≤max{n 1, n 2}－2 and such a drawing can be found in linear time. Add n 1－n 2 dummy vertices to G 2 so that the resulting graph has exactly n 1 vertices. u 2 s s G 2 t G n=12 t u 1 t G 1 n 1=9 n 2=5 =n 1=9 G 2

29 Theorem 1. Let G be a planar graph, and let {G 1, G 2} be an arbitrary bipartition of G. Then G has a grid drawing such that W≤max{n 1, n 2}－1, H≤max{n 1, n 2}－2 and such a drawing can be found in linear time. Add dummy edges to G 2 so that the resulting graph is maximal planar and has an edge (s, t). s G 2 t G n=12 s t n 2=n 1=9 G 2

30 Theorem 1. Let G be a planar graph, and let {G 1, G 2} be an arbitrary bipartition of G. Then G has a grid drawing such that W≤max{n 1, n 2}－1, H≤max{n 1, n 2}－2 and such a drawing can be found in linear time. Embed G 2 so that the edge (s, t) lies on the outer face of G 2. s G 2 t G n=12 s t n 2=n 1=9 G 2

31 Theorem 1. Let G be a planar graph, and let {G 1, G 2} be an arbitrary bipartition of G. Then G has a grid drawing such that W≤max{n 1, n 2}－1, H≤max{n 1, n 2}－2 and such a drawing can be found in linear time. Embed G 2 so that the edge (s, t) lies on the outer face of G 2. s G 2 t G n=12 u 2 s s t t n 2=n 1=9 G 2

32 Theorem 1. Let G be a planar graph, and let {G 1, G 2} be an arbitrary bipartition of G. Then G has a grid drawing such that W≤max{n 1, n 2}－1, H≤max{n 1, n 2}－2 and such a drawing can be found in linear time. Obtain a grid drawing of G 2[CK 97]. Edge (u 2, s) is horizontal. s G 2 t G n=12 t u 2 s t n 2=n 1=9 G 2 u 2 s

Theorem 1 s u 1 t s u 1 u 2 s G 1 G 2 n 1=9 n 2=9 t u 2 s t 33 t

Theorem 1 G s Combine the two drawings and Erase all the dummy vertices and edges. u 2 u 1 34 t n=12 s u 2 s G 2 n 2=9 G 1 n 1=9 u 1 t t

Theorem 1 G s Combine the two drawings and Erase all the dummy vertices and edges. u 2 u 1 35 t n=12 u 2 s G 2 n 2=9 G 1 n 1=9 u 1 t

36 Theorem 1. Let G be a planar graph, and let {G 1, G 2} be an arbitrary bipartition of G. Then G has a grid drawing such that W≤max{n 1, n 2}－1, H≤max{n 1, n 2}－2 and such a drawing can be found in linear time. Such a drawing can be found in linear time, because drawings of G 1, G 2 can be found in linear time by the algorithm in CK 97. W 2=n 1－2 n 1≥n 2 s G 2 s H 1=n 1－2 u 1 t G 1 t W 1=n 1－2 s u 2 H 2=n 1－2 u 1 G 2 u 2 t W=W 1+1 =n 1－1 =max{n 1, n 2}－1 H=H 1 =n 1－2 =max{n 1, n 2}－2 Q. E. D.

37 Our results Theorem 1 If a planar graph G has a balanced bipartition, then G has a grid drawing with small area. This graph has NO balanced bipartition. Lemma 1 Every Series-Parallel graph has a balanced bipartition (n 1, n 2≤ 2 n/3). Planar graph Series-Parallel graph Theorem 2 Series-Parallel W graph H

38 Our results Theorem 1 If a planar graph G has a balanced bipartition, then G has a grid drawing with small area. This graph has NO balanced bipartition. Lemma 1 Every Series-Parallel graph has a balanced bipartition (n 1, n 2≤ 2 n/3). Planar graph Series-Parallel graph Theorem 2 Series-Parallel W graph H

39 Series-Parallel graph A Series-Parallel graph is recursively defined as follows: (1) (2) A single edge is a SP graph． G 2 G 1 terminal : SP graph Series connection G 1 SP graph G 2 G 1 G 2 Parallel connection SP graph

40 Series-Parallel graphs These graphs are Series-Parallel. s t

Bipartition of Series-Parallel graph 41 Lemma 1. Every Series-Parallel graph G of n vertices has a bipartition {G 1, G 2} such that n 1, n 2. Furthermore such a bipartition can be found in linear time. s G G 2 G 1 t SP graph G Bipartition in liner time s s G 1 G 2 t t Subgraph G 1, G 2 n 1, n 2 Suppose for a contradiction that a SP graph has no desired bipartition.

Bipartition of Series-Parallel graph Lemma 1. Every Series-Parallel graph G of n vertices has a bipartition {G 1, G 2} such that n 1, n 2. Furthermore such a bipartition can be found in linear time. Let {s, t} be the most balanced separation pair of G : max{n 1, n 2} is minimum among all bipartitions of G. n 1>2 n/3 G 11 s G 11 G 12 G 1=G 11・G 12 u G 12 G 2 t n 2<n/3 2 -connected SP graph G Assume w. l. o. g. that n 1≥n 2． 42

Bipartition of Series-Parallel graph 43 Lemma 1. Every Series-Parallel graph G of n vertices has a bipartition {G 1, G 2} such that n 1, n 2. Furthermore such a bipartition can be found in linear time. Let {s, t} be the most balanced separation pair of G : max{n 1, n 2} is minimum among all bipartitions of G. n 1>2 n/3 G 1=G 11・G 12 Assume w. l. o. g. that n 1≥n 2 and n 11≥n 12. G 1 n 11>n/3 G 11 n 1> n 11 u G 12 n 1> n 11 G s t 11 n 1> max{n 11, n 11} G 2 Contradiction. n 11<2 n/3 n 1=max{n 1, n 2} n 2<n/3 max{n 1, n 2}> max{n 11, n 11} 2 -connected SP graph G

Grid drawing of Series-Parallel graph s G G 2 G 1 t SP graph G Lemma 1 in linear time s s G 2 G 1 t t Subgraph G 1, G 2 n 1, n 2 44

Grid drawing of Series-Parallel graph s G G 1 t s G 2 t t t Subgraph G 1, G 2 n 1, n 2 s G 1 G 2 G 1 SP graph G s s s Lemma 1 in linear time G 2 45 t Subgraph G 1, G 2 Theorem 1 in linear time u 1 G 2 u 2 H≤max{n 1, n 2}－2 t W≤max{n 1, n 2}－1

Grid drawing of Series-Parallel graph s G G 1 t t Subgraph G 1, G 2 n 1, n 2 s s G 2 t G 2 G 1 SP graph G s s s Lemma 1 in linear time G 2 46 t SP Subgraph G 1, G 2 n 1, n 2 Theorem 1 in linear time u 1 G 2 u 2 H≤max{n 1, n 2}－2 t W≤max{n 1, n 2}－1

Grid drawing of Series-Parallel graph 47 Theorem 2. Every Series-Parallel graph of n vertices has a grid drawing such that W H. Furthermore such a drawing can be found in linear time. s G 1 s s G 2 t t SP Subgraph G 1, G 2 n 1, n 2 Theorem 1 in linear time u 1 G 2 u 2 H≤max{n 1, n 2}－2 t W≤max{n 1, n 2}－1

48 Grid drawing of Series-Parallel graph u 2 s s H=7 Theorem 2 in linear time t n=12 in Lem lin m ea a 1 rt im e SP graph G s u 1 s G 1 n 1=9 t G 2 t n 2=5 W=8 1 m re time o e ar h T ne li n i t

49 Conclusions Gird drawing Width and Height Running time W≤max{n 1, n 2}－1 Planar graph with balanced bipartition s u 1 u 2 G 1 H≤max{n 1, n 2}－2 O(n) t W s SP graph Partial 2 -tree u 1 G 2 t u 2 O(n) H

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