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GSIS, Tohoku University 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 3

Book 4

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

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 2 3 3 4 4 1 1 5 5 6 6 7 7

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

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

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

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

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 u2 Bipartition Drawing s s G2 s G2 G2 t G1 G1 G1 G t u1 t t Planar graph G Subgraph G1,G2 Drawing of G

(3) Maximal planar graph G1,G2 Outline of algorithm Maximal planar graph Bipartition s s G2 G2 s G1 t u2 G1 s G t t G2 s (1) Planar graph G (2) Subgraph G1,G2 t G1 u1 t (3) Maximal planar graph G1,G2 u2 u2 Drawing s Combining s s G2 G2 G1 G1 u1 u1 t t t (5) Drawing of G (4) Drawing of G1,G2

Our results s G2 Theorem 1 G1 If a planar graph G has a balanced bipartition, then G has a grid drawing with small area. G t (1) Planar graph G G1,G2 Width, Height n1,n2≤n/2 W,H≤n/2 W≤max{n1, n2}－1 u2 s n1,n2≤2n/3 W,H≤2n/3 G2 H≤max{n1, n2}－2 G1 n1,n2≤αn W,H≤αn u1 t If α<1, Balanced bipartition (5) Drawing of G

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. G1,G2 Width, Height n1,n2≤n/2 W,H≤n/2 n1,n2≤2n/3 W,H≤2n/3 n1,n2≤αn W,H≤αn If α<1, Balanced bipartition

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 (n1,n2≤2n/3). Planar graph α=2/3 Series-Parallel graph

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 (n1,n2≤2n/3). Planar graph α=2/3 Series-Parallel graph Theorem 2 Series-Parallel graph W H

(3) Maximal planar graph G1,G2 Outline of algorithm Maximal planar graph Bipartition s s G2 G2 s G1 t u2 G1 s G t t G2 s (1) Planar graph G (2) Subgraph G1,G2 t G1 u1 t (3) Maximal planar graph G1,G2 u2 u2 Drawing s Combining s s G2 G2 G1 G1 u1 u1 t t t (5) Drawing of G (4) Drawing of G1,G2

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 G1=(V1,E1) and G2=(V2,E2) such that 　・ V=V1∪V2，V1∩V2={s,t}, 　・ E=E1∪E2，E1∩E2=∅． Such a pair of subgraphs {G1,G2} is called a bipartition of G. G2 s s s Bipartition n1=9 t t t G1 n2=5 n=12 (1) Graph G (2) Subgraph G1,G2

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 G1=(V1,E1) and G2=(V2,E2) such that 　・ V=V1∪V2，V1∩V2={s,t}, 　・ E=E1∪E2，E1∩E2=∅． Such a pair of subgraphs {G1,G2} is called a bipartition of G. G2 t t t G1 Bipartition s s s n1=3 n=12 n2=11 (1) Graph G (2) Subgraph G1,G2

(3) Maximal planar graph G1,G2 Outline of algorithm Maximal planar graph Bipartition s s G2 G2 s G1 t u2 G1 s G t t G2 s (1) Planar graph G (2) Subgraph G1,G2 t G1 u1 t (3) Maximal planar graph G1,G2 u2 u2 Drawing s Combining s s G2 G2 G1 G1 u1 u1 t t t (5) Drawing of G (4) Drawing of G1,G2

Theorem 1 Theorem 1. Let G be a planar graph, and let {G1,G2} be an arbitrary bipartition of G. Then G has a grid drawing such that W≤max{n1,n2}－1, H≤max{n1,n2}－2 and such a drawing can be found in linear time. W≤max{n1, n2}－1 Drawing in linear time u2 s s G2 G2 H≤max{n1, n2}－2 G1 G1 G t u1 t Planar graph G Drawing of G

Theorem 1 Theorem 1. Let G be a planar graph, and let {G1,G2} be an arbitrary bipartition of G. Then G has a grid drawing such that W≤max{n1,n2}－1, H≤max{n1,n2}－2 and such a drawing can be found in linear time. Assume w.l.o.g. that n1≥n2． s s s G1 t t t G2 G1 n2=5 G n=12 n1=9

Theorem 1 Theorem 1. Let G be a planar graph, and let {G1,G2} be an arbitrary bipartition of G. Then G has a grid drawing such that W≤max{n1,n2}－1, H≤max{n1,n2}－2 and such a drawing can be found in linear time. Assume w.l.o.g. that n1≥n2． Add dummy edges to G1 so that the resulting graph is maximal planar and has an edge (s,t). s s G1 t t G1 G n=12 n1=9

Theorem 1 Theorem 1. Let G be a planar graph, and let {G1,G2} be an arbitrary bipartition of G. Then G has a grid drawing such that W≤max{n1,n2}－1, H≤max{n1,n2}－2 and such a drawing can be found in linear time. Embed G1 so that the edge (s,t) lies on the outer face of G1. s s G1 t u1 t G1 G n=12 n1=9

Theorem 1 Theorem 1. Let G be a planar graph, and let {G1,G2} be an arbitrary bipartition of G. Then G has a grid drawing such that W≤max{n1,n2}－1, H≤max{n1,n2}－2 and such a drawing can be found in linear time. Obtain a grid drawing of G1[CK97]. s s G1 t u1 t G1 G n=12 n1=9

Theorem 1 Theorem 1. Let G be a planar graph, and let {G1,G2} be an arbitrary bipartition of G. Then G has a grid drawing such that W≤max{n1,n2}－1, H≤max{n1,n2}－2 and such a drawing can be found in linear time. s Obtain a grid drawing of G1[CK97]. u1 t G1 s n1=9 H1=n1－2 Edge (u1,t) is horizontal. G1 n1=9 u1 t W1=n1－2

Theorem 1 Theorem 1. Let G be a planar graph, and let {G1,G2} be an arbitrary bipartition of G. Then G has a grid drawing such that W≤max{n1,n2}－1, H≤max{n1,n2}－2 and such a drawing can be found in linear time. u2 s G2 s s G2 t t G2 u1 n2=5 t G n=12 G1 n1=9

Theorem 1 Theorem 1. Let G be a planar graph, and let {G1,G2} be an arbitrary bipartition of G. Then G has a grid drawing such that W≤max{n1,n2}－1, H≤max{n1,n2}－2 and such a drawing can be found in linear time. Add n1－n2 dummy vertices to G2 so that the resulting graph has exactly n1 vertices. u2 s G2 s s G2 t t G2 u1 n2=5 n2=n1=9 t G n=12 G1 n1=9

Theorem 1 Theorem 1. Let G be a planar graph, and let {G1,G2} be an arbitrary bipartition of G. Then G has a grid drawing such that W≤max{n1,n2}－1, H≤max{n1,n2}－2 and such a drawing can be found in linear time. Add dummy edges to G2 so that the resulting graph is maximal planar and has an edge (s,t). G2 s s t t G2 n2=n1=9 G n=12

Theorem 1 Theorem 1. Let G be a planar graph, and let {G1,G2} be an arbitrary bipartition of G. Then G has a grid drawing such that W≤max{n1,n2}－1, H≤max{n1,n2}－2 and such a drawing can be found in linear time. Embed G2 so that the edge (s,t) lies on the outer face of G2. G2 s s t t G2 n2=n1=9 G n=12

Theorem 1 Theorem 1. Let G be a planar graph, and let {G1,G2} be an arbitrary bipartition of G. Then G has a grid drawing such that W≤max{n1,n2}－1, H≤max{n1,n2}－2 and such a drawing can be found in linear time. Embed G2 so that the edge (s,t) lies on the outer face of G2. u2 G2 s s s t t t G2 G2 n2=n1=9 n2=n1=9 G n=12

Theorem 1 Theorem 1. Let G be a planar graph, and let {G1,G2} be an arbitrary bipartition of G. Then G has a grid drawing such that W≤max{n1,n2}－1, H≤max{n1,n2}－2 and such a drawing can be found in linear time. Obtain a grid drawing of G2[CK97]. Edge (u2,s) is horizontal. u2 G2 t s u2 s s t t G2 n2=n1=9 G n=12

Theorem 1 s u2 s u1 t t G1 G2 n1=9 n2=9 u2 s s u1 t t

Theorem 1 G s u2 Combine the two drawings and Erase all the dummy vertices and edges. u1 t n=12 t s u2 s G2 n2=9 G1 n1=9 u1 t

Theorem 1 G s u2 Combine the two drawings and Erase all the dummy vertices and edges. u1 t n=12 u2 s G2 n2=9 G1 n1=9 u1 t

Theorem 1 Theorem 1. Let G be a planar graph, and let {G1,G2} be an arbitrary bipartition of G. Then G has a grid drawing such that W≤max{n1,n2}－1, H≤max{n1,n2}－2 and such a drawing can be found in linear time. Such a drawing can be found in linear time, because drawings of G1,G2 can be found in linear time by the algorithm in CK97. n1≥n2 W2=n1－2 u2 s u2 s G2 H=H1 =n1－2 =max{n1, n2}－2 G2 s H2=n1－2 G1 u1 t H1=n1－2 t G1 u1 t W=W1+1 =n1－1 =max{n1, n2}－1 Q.E.D. W1=n1－2

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 (n1,n2≤2n/3). Planar graph Series-Parallel graph Theorem 2 Series-Parallel graph W H

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 (n1,n2≤2n/3). Planar graph Series-Parallel graph Theorem 2 Series-Parallel graph W H

Series-Parallel graph A Series-Parallel graph is recursively defined as follows: (1) A single edge is a SP graph． terminal (2) (2) : SP graph G1 G1 G2 G2 Series connection SP graph G1 G2 G1 Parallel connection SP graph G2

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

Bipartition of Series-Parallel graph Lemma 1. Every Series-Parallel graph G of n vertices has a bipartition {G1,G2} such that n1,n2 . Furthermore such a bipartition can be found in linear time. Bipartition in liner time s s s G2 G2 G1 t G1 G t t SP graph G Subgraph G1,G2 n1,n2 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 {G1,G2} such that n1,n2 . Furthermore such a bipartition can be found in linear time. Let {s,t} be the most balanced separation pair of G : max{n1,n2} is minimum among all bipartitions of G. n1>2n/3 G1=G11・G12 Assume w.l.o.g. that n1≥n2． G1 G11 G11 u G12 G12 s t G2 n2<n/3 2-connected SP graph G

Bipartition of Series-Parallel graph Lemma 1. Every Series-Parallel graph G of n vertices has a bipartition {G1,G2} such that n1,n2 . Furthermore such a bipartition can be found in linear time. Let {s,t} be the most balanced separation pair of G : max{n1,n2} is minimum among all bipartitions of G. n1>2n/3 G1=G11・G12 Assume w.l.o.g. that n1≥n2 and n11≥n12. n11>n/3 G1 G11 n1> n11 u G12 n1> n11 G11 s t G2 n1> max{n11,n11} n11<2n/3 Contradiction. n1=max{n1,n2} n2<n/3 max{n1,n2}> max{n11,n11} 2-connected SP graph G

Grid drawing of Series-Parallel graph Lemma 1 in linear time s s s G2 G2 G1 G1 t G t t SP graph G Subgraph G1,G2 n1,n2

Grid drawing of Series-Parallel graph Lemma 1 in linear time s s s G2 G2 G1 G1 t G t t SP graph G Subgraph G1,G2 n1,n2 u2 s s Theorem 1 in linear time s G2 G2 H≤max{n1, n2}－2 G1 G1 u1 t t t Subgraph G1,G2 W≤max{n1, n2}－1

Grid drawing of Series-Parallel graph Lemma 1 in linear time s s s G2 G2 G1 G1 t G t t SP graph G Subgraph G1,G2 n1,n2 u2 s s Theorem 1 in linear time s G2 G2 H≤max{n1, n2}－2 G1 G1 u1 t t t SP Subgraph G1,G2 W≤max{n1, n2}－1 n1,n2

Grid drawing of Series-Parallel graph 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. u2 s s Theorem 1 in linear time s G2 G2 H≤max{n1, n2}－2 G1 G1 u1 t t t SP Subgraph G1,G2 W≤max{n1, n2}－1 n1,n2

Grid drawing of Series-Parallel graph u2 s s Theorem 2 in linear time H=7 t SP graph G n=12 u1 t W=8 Lemma 1 in linear time s s Theorem 1 in linear time G1 n1=9 t G2 t n2=5

with balanced bipartition Conclusions Gird drawing Width and Height Running time Planar graph with balanced bipartition O(n) SP graph Partial 2-tree W≤max{n1, n2}－1 u2 s G2 H≤max{n1, n2}－2 G1 u1 t W u2 s G2 G1 H u1 t