Minimum-Segment Convex Drawings of 3-Connected Cubic Plane Graphs Sudip Biswas Debajyoti Mondal Rahnuma Islam Nishat Md. Saidur Rahman Graph Drawing and Information Visualization Laboratory Department of Computer Science and Engineering Bangladesh University of Engineering and Technology (BUET) Dhaka – 1000, Bangladesh COCOON 2010July 19, 2010

Minimum-Segment Convex Drawings Convex Drawing

9 segments 8 segments 6 segments Minimum-Segment Convex Drawings Minimum-Segment

Previous Results M. Chrobak et al. [1997] Straight-line convex grid drawings of 3-connected plane graphs (n-2) x (n-2) area G. Kant [1994] Orthogonal grid drawings of 3-connected cubic plane graphs ( n / 2 +1)x( n / 2 +1) area Dujmovic et al. [2006] Straight-line drawings of cubic graphs with few segments (n-2) segments Keszegh et al. [2008] Straight-line drawings with few slopes 5 slopes and at most 3 bends

Our Results Straight-line convex grid-drawings of cubic graphs ( n / 2 +1) x ( n / 2 +1) area Minimum segment 6 slopes, no bend

Straight-line convex grid-drawings of cubic graphs Input: 3-Connected Plane Cubic Graph G Output: Minimum- Segment Drawing of G

Intuitive Idea A Minimum-Segment Drawing Vertices on the same segment have straight corners

Intuitive Idea A Minimum-Segment Drawing number of segment decreases after ensuring a straight corner at a vertex Lets try to ensure a straight corner at each vertex in the drawing

An Example … How do we choose the set of vertices at each step? The number of straight corners is (n-3) and this is the maximum The number of segments is the minimum.

An Example Canonical Decomposition G. Kant: Every 3-connected plane graph has a canonical decomposition which can be obtained in linear time … Choose a partition at each step such that the resulting graph is 2-connected

Let’s Impose some rules Chain 3 is the left-end of the chain {7,8} 4 is the right-end of the chain {7,8} (3,7) is the left-edge of {7,8} (4,8) is the right-edge of {7,8}

Let’s Impose some rules If the left-end of the chain has a straight corner, use slope +1

Let’s Impose some rules If the right-end of the chain has a straight corner, use slope -1

Let’s Impose some rules If the right-end is at the rightmost position of the drawing, use the slope 

Let’s Impose some rules In all other cases, use the slope of the outer-edges. Slope of (7,8) = Slope of (8,11)

If the right-end of the chain has a straight corner, use slope -1 If the left-end of the chain has a straight corner, use slope +1 If the right-end is at the rightmost position of the drawing, use the slope  In all other cases, use the slope of the outer-edges. These four rules works for minimum-segment convex drawings! Minimum-Segment Convex Drawings

How can we obtain a grid drawing? Minimum-Segment Convex Drawings

Minimum-Segment Convex Grid Drawings Now the rules of placing the partitions are not so simple!

An Example …

Calculation of Grid Size … |V 1 | = 6 Width= 6 |V 2 | = 2 Width= 6+1= 7 Width= |V 1 | + (|V 2 |-1) |V 3 | = 2 Width= 7+1=8 Width= |V 1 | + (|V 2 |-1) + (|V 3 |-1) Width = |V 1 | + ∑ (|V k |-1) = |V 1 | + ∑ (|V k |-1) = n -∑ k 1 = n / 2 +1

Calculation of Grid Size … n/2n/2 n/2n/2 Area of the drawing = ( n / 2 +1) x ( n / 2 +1)

The number of slopes is six … o0o 45 o  (1,14) (1, 6) (5, 6)

Thank You

An Example

An Example

An Example

An Example

An Example

Canonical decomposition of a 3-Connected Cubic graph G. Kant: Every 3-connected plane graph has a canonical decomposition which can be obtained in linear time G1G1 G2G2 G3G3 G4G4 G5G5 G k-1 GkGk

Canonical Ordering of a 3-Connected Cubic graph We call a partition V k = (z 1, z 2,…, z l ) The leftmost and the rightmost neighbor of V k on G k-1 are w p and w q wpwp wqwq z1z1 z2z2 z3z3 z4z4 G k-1 GkGk

Minimum-Segment Convex Drawing G 1 is drawn as a tricorner. (z 1, z 2,…, z l ) form a segment of slope A Vertex with a Straight Corner (180 o ) w1w1 w2w2 w3w3 w4w

V k = (6, 5), w p =3, w q = If x(w q ) is the maximum, then slope(w q, z l ) = ∞ If x(w p ) is not the minimum, then slope(w p, w p-1 ) = slope(w p, z 1 ) w p-1 wpwp z1z1 wqwq zlzl Minimum-Segment Convex Drawing

V k = (10,9,8,7), w p =4, w q =6. If w q has a straight corner, then slope(w q, z 1 ) = -1 If w p has a straight corner, then slope(w p, z 1 ) = +1 w p-1 z1z1 zlzl wpwp wqwq Minimum-Segment Convex Drawing

V k = (11), w p =1, w q =10. If x(w q ) is not the minimum, then slope(w q, w q+1 ) = slope(w q, z l ) If x(w p ) is the minimum, then slope(w p, z 1 ) = +1 z l = z l wpwp wqwq w q+1 Minimum-Segment Convex Drawing

V k = (11), w p =1, w q =10. If x(w q ) is not the minimum, then slope(w q, w q+1 ) = slope(w q, z l ) If x(w p ) is the minimum, then slope(w p, z 1 ) = +1 z l = z l wpwp wqwq w q+1 Minimum-Segment Convex Drawing

V k = (13,12), w p =7 w q = V k = (15,14), w p =9, w q = V k = (16), w p =11, w q = V k = (17), w p =13, w q = V k = (18), w p =16, w q =17. Minimum-Segment Convex Drawing

The vertices 1, 2, 14 and 18 do not have any straight corner Minimum-Segment Convex Drawing 14 The vertices 1, 2 and 18 do not have any straight corner

13 A segment S has |S| edges and |S|-1 straight corners. Minimum-Segment Convex Drawing S |S| = 6 Number of Straight Corners =|S| - 1

A segment S has |S| edges and |S|-1 straight corners. Minimum-Segment Convex Drawing Γ Let, Γ has n>4 vertices and x segments. Denote the segments by S 1, S 2, …, S x x is the minimum (|S 1 |-1) + (|S 2 |-1)+ … + (|S x |-1) = (n-3) (|S 1 | + |S 2 |+ … + |S x |) - x = (n-3) Constant - x = (n-3) x = Constant - (n-3)