Drawing Plane Graphs. Takao Nishizeki. Tohoku University. US President. California Governor. US President. California Governor. What is the common feature?. STATION. STATION. STATION. STATION. STATION. ATM-HUB. ATM-RT. ATM-RT. ATM-SW. STATION. STATION. TPDDI. ATM-HUB. ATM-SW.

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Drawing Plane Graphs

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US President California Governor What is the common feature?

STATION STATION STATION STATION STATION ATM-HUB ATM-RT ATM-RT ATM-SW STATION STATION TPDDI ATM-HUB ATM-SW ATM-RT ATM-SW STATION STATION ATM-HUB STATION ATM-SW TPDDI ATM-SW STATION TPDDI STATION ATM-HUB ATM-HUB ATM-RT STATION STATION STATION STATION STATION STATION Graphs and Graph Drawings A diagram of a computer network

Symmetric Eades, Hong Objectives of Graph Drawings Nice drawing structure of the graph is easy to understand structure of the graph is difficult to understand To obtain a nice representation of a graph so that the structure of the graph is easily understandable.

Modern beauty Objectives of Graph Drawings Nice drawing Ancient beauty

8 7 5 6 4 3 2 1 Objectives of Graph Drawings Diagram of an electronic circuit 5 Wire crossings 7 4 8 3 1 2 not suitable for single layered PCB suitable for single layered PCB The drawing should satisfy some criterion arising from the application point of view.

Straight Line Drawing Straight line drawing Plane graph Each vertex is drawn as a point.

Straight Line Drawing Straight line drawing Plane graph Each vertex is drawn as a point. Each edge is drawn as a single straight line segment.

Every plane graph has a straight line drawing. Wagner ’36 Fary ’48 Straight Line Drawing Polynomial-time algorithm Straight line drawing Plane graph Each vertex is drawn as a point. Each edge is drawn as a single straight line segment.

W H Straight Line Drawing W H Straight line drawing Plane graph

Straight Line Grid Drawing Straight line grid drawing. Plane graph In a straight line grid drawing each vertex is drawn on a grid point.

Wagner ’36 Fary ’48 Grid-size is not polynomial of the number of vertices n Straight line grid drawing. Plane graph

Triangulate all inner faces Step1: find a 4-canonical ordering G” n=18 n-1=17 n=18 n-1=17 16 16 Step2: Divide G into two halvesG’ and G” 15 15 12 12 14 14 10 10 13 13 11 11 9 n/2=9 8 8 5 5 6 6 7 7 4 4 Step3 and 4 : Draw G’ and G” in isosceles right-angled triangles 3 3 1 2 1 2 G’ G” W/2 G 1 |slope|>1 |slope|>1 |slope|>1 Step5: Combine the drawings of G’ and G” n/2 W/2 |slope|<1 = G’ W < n/2 -1 n/2 -1 = Main idea G

Draw a graph G on the plane “nicely” Straight line drawing A convex drawing is a straight line drawing where each face is drawn as a convex polygon. Convex drawing

Convex Drawing Convex drawing Tutte 1963 Every 3-connected planar graph has a convex drawing. A necessary and sufficient condition for a plane graph to have a convex drawing. Thomassen ’80

Convex Drawing Chiba et al. ’84 O(n) time algorithm Convex drawing Tutte 1963 Every 3-connected planar graph has a convex drawing A necessary and sufficient condition for a plane graph to have a convex drawing. Thomassen ’80

Convex Grid Drawing Chrobak and Kant ’97 Input: 3-connected graph Output: convex grid drawing n-2 n-2 Grid Size Area

Convex Grid Drawing Miura et al. 2000 Input : 4-connected plane graph Output: Convex grid drawing H Grid Size W Half-perimeter Area

Area < 1/4 = Miura et al. 2000 Chrobak and Kant ’97 3-connected graph 4-connected graph n-2 H n-2 W Area Area

VLSI Floorplanning B B A A F F E E C G C G D D VLSI floorplan Interconnection graph

VLSI Floorplanning B B A A F F E E C G C G D D VLSI floorplan Interconnection graph

VLSI Floorplanning B B A A F F E E C G C G D D VLSI floorplan Interconnection graph

B A F E G C D VLSI Floorplanning B B A A F F E E C G C G D D VLSI floorplan Interconnection graph Dual-like graph

B B A A F F E E G G C C D D VLSI Floorplanning B B A A F F E E C G C G D D VLSI floorplan Interconnection graph Dual-like graph Add four corners

B B A A F F E E G G C C D D VLSI Floorplanning B B A A F F E Rectangular drawing E C G C G D D VLSI floorplan Interconnection graph Dual-like graph Add four corners

Rectangular Drawings corner Rectangular drawing of G Plane graph G of Output Input Each vertex is drawn as a point.

Rectangular Drawings corner Rectangular drawing of G Plane graph G of Output Input Each vertex is drawn as a point. Each edge is drawn as a horizontal or a vertical line segment.

Rectangular Drawings corner Rectangular drawing of G Plane graph G of Output Input Each vertex is drawn as a point. Each edge is drawn as a horizontal or a vertical line segment. Each face is drawn as a rectangle.