Vertex orderings Vertex ordering 16 13 12 11

Vertex ordering 16 13 12 11 9 8 7 14 10 6 5 2

st-numbering t = 16 13 12 11 9 8 7 has two neighbors j,

st-numbering t = 16 13 12 11 9 8 7 14 10 6 5

Application of st-numbersing Planarity testing Visibility drawing Internet routing

Canonical Ordering 16 13 15 10 9 8 Triangulated plane graph 1 14 12

Canonical Ordering 16 13 15 10 9 8 14 12 7 1 Gk :

Canonical Ordering 16 13 15 10 14 12 11 G 9 9 8 7

Canonical Ordering For any 16 13 15 10 (co 1) Gk is biconnected and

Every triangulated plane graph has a canonical ordering.

Canonical Ordering Chord: For a cycle C in a graph, an edge joining two

Lemma 4. 2. 1 Every triangulated plane graph G has a canonical ordering. Proof.

w = vk Gk-1 v 2 Now we have to proof that there is

In Case (i), any of the vertices w 2, w 3, …, wt-1 is

Algorithm: Canonical-Ordering In the animation, Red number means ordering; Outer vertex are colored red;

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

Every plane graph has a straight line Straight Line Drawing drawing. Wagner ’ 36

Straight Line Grid Drawing Straight line grid drawing. Plane graph In a straight line

Every plane graph has a straight line Straight Line Grid Drawing drawing. Wagner ’

Straight Line Grid Drawing W H Straight line grid drawing. Plane graph de Fraysseix

What is the minimum size of a grid required for a straight line drawing?

de Fraysseix ‘ 90 Schnyder ‘ 90 Shift Method Realizer Method

Shift Method 16 13 15 14 10 9 6 8 7 1 12 11

Some other vertex orsering • Antibandwidth labeling • Graceful labeling

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Vertex orderings

Vertex ordering 16 13 12 11 9 8 7 14 10 6 5 2 1 15 4 3

st-numbering t = 16 13 12 11 9 8 7 has two neighbors j, k 14 10 6 5 2 s=1 15 4 3

st-numbering t = 16 13 12 11 9 8 7 14 10 6 5 2 s=1 and For any i, both vertices induce connected subgraphs. 15 4 3

Application of st-numbersing Planarity testing Visibility drawing Internet routing

Canonical Ordering 16 13 15 10 9 8 Triangulated plane graph 1 14 12 7 11 4 6 5 3 2

Canonical Ordering 16 13 15 10 9 8 14 12 7 1 Gk : subgraph of G induced by vertices 11 4 6 5 3 2

Canonical Ordering 16 13 15 10 14 12 11 G 9 9 8 7 1 Gk : subgraph of G induced by vertices 4 6 5 3 2

Canonical Ordering For any 16 13 15 10 (co 1) Gk is biconnected and internally triangulated 9 8 14 12 7 11 4 6 5 3 1 (co 2) vertices 1 and 2 are on the outer face of Gk (co 3) vertex k+1 is on the outer face of Gk and the neighbor of k+1 is consecutive on the outer cycle Co(Gk). 2

Canonical Ordering For any 16 13 15 10 (co 1) Gk is biconnected and internally triangulated 9 8 1 14 12 7 (co 2) vertices 1 and 2 are on the outer face of Gk 11 4 6 5 3 G 3 (co 3) vertex k+1 is on the outer face of Gk and the neighbor of k+1 is consecutive on the outer cycle Co(Gk). 2

Canonical Ordering For any 16 13 15 10 (co 1) Gk is biconnected and internally triangulated 9 8 1 14 12 7 (co 2) vertices 1 and 2 are on the outer face of Gk 11 4 6 5 3 G 4 (co 3) vertex k+1 is on the outer face of Gk and the neighbor of k+1 is consecutive on the outer cycle Co(Gk). 2

Canonical Ordering For any 16 13 15 10 (co 1) Gk is biconnected and internally triangulated 9 8 1 14 12 7 (co 2) vertices 1 and 2 are on the outer face of Gk 11 4 6 5 3 G 10 (co 3) vertex k+1 is on the outer face of Gk and the neighbor of k+1 is consecutive on the outer cycle Co(Gk). 2

Every triangulated plane graph has a canonical ordering.

Canonical Ordering Chord: For a cycle C in a graph, an edge joining two non-consecutive vertices in C is called a chord of C.

Lemma 4. 2. 1 Every triangulated plane graph G has a canonical ordering. Proof. Using reverse induction Basis: Since G = Gn, clearly (co 1)-(co 3) hold for k=n. Inductive hypothesis: The vertices vn, vn-1, …, vk+1, k+1≥ 4 have been appropriately chosen. Induction step: Now we have to choose vk. If one can choose as vk a vertex w ≠ v 1, v 2 on the cycle Co(Gk) which is not an end of a chord of Co(Gk) then clearly (co 1)-(co 3) hold for k-1 since Gk-1 = Gk – vk.

w = vk Gk-1 v 2 Now we have to proof that there is such a vertex. We have to consider two cases – (i) Co(Gk) has no chord (ii) Co(Gk) has chord. Let Co(Gk) = w 1, w 2, …. , wt, where w 1= v 1 and wt = v 2.

In Case (i), any of the vertices w 2, w 3, …, wt-1 is such a vertex w. wp+1 wp wq-1 wq w 2 w 1=v 1 wt=v 2 In case (ii), find a minimal chord (wp, wq), p+2≤q, and any of the vertices wp+1, wp+2, …, wq-1 is such a vertex w.

Algorithm: Canonical-Ordering In the animation, Red number means ordering; Outer vertex are colored red; Blue number indicates number of chords of the outer cycles end with the associated vertex

16 0 15 0 9 1 0 v 1 13 0 0 10 08 0 0 7 14 0 0 C 0(G 14) 0 12 11 0 3 0 4 Co(G 15) 0 0 6 5 C 0(G 13) 2 0 v 2 Choose a vertex x such that chords(x) = 0 and x ≠ v 1, v 2

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 Straight Line Drawing drawing. Wagner ’ 36 Fary ’ 48 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.

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

Every plane graph has a straight line Straight Line Grid Drawing drawing. Wagner ’ 36 Fary ’ 48 Grid-size is not polynomial of the number of vertices n Straight line grid drawing. Plane graph

Straight Line Grid Drawing W H Straight line grid drawing. Plane graph de Fraysseix ‘ 90

Schnyder ‘ 90 H n-2 W

What is the minimum size of a grid required for a straight line drawing?

de Fraysseix ‘ 90 Schnyder ‘ 90 Shift Method Realizer Method

Shift Method 16 13 15 14 10 9 6 8 7 1 12 11 3 4 5 2 Canonically ordered input graph G

16 13 15 14 10 9 1 12 11 3 2 6 8 7 3 4 5 2 1 3 1 2 Straight-line grid drawing of G using shift method 1 (0, 0) 2 3 (2, 0) (1, 1)

16 13 15 14 10 9 12 11 3 4 2 6 8 7 5 2 1 3 1 1 4 2 Straight-line grid drawing of G using shift method 1 (0, 0) 2 3 4 (4, 0) (1, 1) (2, 2)

16 13 15 14 10 9 12 11 3 4 5 6 8 7 2 5 2 1 3 1 1 4 5 2 Straight-line grid drawing of G using shift method 1 (0, 0) 2 3 4 5 (6, 0) (1, 1) (2, 2) (3, 3)

16 13 15 10 9 1 14 3 12 11 4 7 3 4 6 5 2 2 1 3 1 5 6 8 4 5 6 2 Straight-line grid drawing of G using shift method 1 (0, 0) 2 3 4 5 6 (8, 0) (1, 1) (2, 2) (3, 3) (4, 4)

16 13 15 10 9 1 14 7 12 11 4 7 3 4 6 5 2 2 1 5 7 4 3 1 5 6 8 3 6 2 Straight-line grid drawing of G using shift method 1 (0, 0) 2 3 4 5 6 7 (10, 0) (3, 1) (4, 2) (5, 3) (6, 4) (2, 2)

16 1 13 15 10 9 7 12 11 4 3 3 4 5 6 8 7 6 5 2 2 1 8 14 5 74 3 6 2 Straight-line grid drawing of G using shift method 1 (0, 0) 2 3 4 5 6 7 8 (12, 0) (5, 1) (6, 2) (7, 3) (8, 4) (4, 2) (3, 3)

1 16 9 13 15 14 10 9 8 7 12 11 7 3 4 5 5 6 2 2 1 9 1 4 6 8 8 5 7 4 3 3 6 2 Straight-line grid drawing of G using shift method

1 16 9 13 15 10 9 10 14 8 12 11 8 7 3 4 3 5 5 6 7 6 4 2 2 1 10 9 1 8 7 45 3 6 2 Straight-line grid drawing of G using shift method

1 16 9 13 15 10 9 10 14 12 11 11 7 3 4 3 5 6 2 2 1 10 9 1 5 7 6 8 4 8 8 11 7 4 3 6 5 2 Straight-line grid drawing of G using shift method

1 16 9 13 15 10 9 10 14 8 12 11 8 7 3 4 5 6 2 2 1 12 10 9 1 3 5 12 11 7 6 4 8 7 4 3 11 5 6 2 Straight-line grid drawing of G using shift method

16 1 13 15 10 9 9 14 10 8 12 11 7 3 4 3 5 13 12 11 7 6 8 4 5 2 2 1 13 12 11 10 9 1 8 7 3 6 4 6 5 2 Straight-line grid drawing of G using shift method

16 1 13 15 10 9 9 14 10 8 12 11 7 6 8 7 3 4 4 3 5 13 12 5 14 11 6 2 1 14 13 10 9 1 8 12 11 7 3 4 6 5 2 Straight-line grid drawing of G using shift method 2

16 1 13 15 14 10 9 9 12 11 15 4 10 8 7 3 4 5 14 12 5 6 8 13 11 7 3 2 1 15 14 13 12 10 9 1 8 7 3 11 4 6 5 2 Straight-line grid drawing of G using shift method 6 2

16 1 13 15 14 10 9 9 12 11 10 8 6 8 7 4 3 5 15 4 13 14 12 5 11 7 3 2 1 16 14 15 13 12 10 9 1 8 7 3 11 4 6 5 2 Straight-line grid drawing of G using shift method 6 2

Some other vertex orsering • Antibandwidth labeling • Graceful labeling