Vertex orderings.

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Presentation transcript:

Vertex orderings

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

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

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

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

Application of st-numbersing Planarity testing Visibility drawing Internet routing

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

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

Canonical Ordering 16 13 14 15 10 12 11 6 G9 5 9 4 8 7 3 1 2 Gk : subgraph of G induced by vertices

Canonical Ordering 16 13 14 For any 15 10 12 11 6 5 9 4 8 7 3 1 2 (co1) Gk is biconnected and internally triangulated 9 4 8 7 3 1 2 (co2) vertices 1 and 2 are on the outer face of Gk (co3) vertex k+1 is on the outer face of Gk and the neighbor of k+1 is consecutive on the outer cycle Co(Gk).

Canonical Ordering 16 13 14 For any 15 10 12 11 6 5 9 4 8 7 3 1 2 G3 (co1) Gk is biconnected and internally triangulated 9 4 8 7 3 1 2 G3 (co2) vertices 1 and 2 are on the outer face of Gk (co3) vertex k+1 is on the outer face of Gk and the neighbor of k+1 is consecutive on the outer cycle Co(Gk).

Canonical Ordering 16 13 14 For any 15 10 12 11 6 5 9 4 8 7 3 1 2 G4 (co1) Gk is biconnected and internally triangulated 9 4 8 7 3 1 2 G4 (co2) vertices 1 and 2 are on the outer face of Gk (co3) vertex k+1 is on the outer face of Gk and the neighbor of k+1 is consecutive on the outer cycle Co(Gk).

Canonical Ordering 16 13 14 For any 15 10 12 11 6 5 9 4 8 7 3 1 2 G10 (co1) Gk is biconnected and internally triangulated 9 4 8 7 3 1 2 G10 (co2) vertices 1 and 2 are on the outer face of Gk (co3) vertex k+1 is on the outer face of Gk and the neighbor of k+1 is consecutive on the outer cycle Co(Gk).

Canonical Ordering 16 13 14 For any 15 10 12 11 6 5 9 4 8 7 3 1 2 G10 (co1) Gk is biconnected and internally triangulated 9 4 8 7 3 1 2 G10 (co2) vertices 1 and 2 are on the outer face of Gk (co3) vertex k+1 is on the outer face of Gk and the neighbor of k+1 is consecutive on the outer cycle Co(Gk).

Canonical Ordering 16 13 14 For any 15 10 12 11 6 5 9 4 8 7 3 1 2 G10 (co1) Gk is biconnected and internally triangulated 9 4 8 7 3 1 2 G10 (co2) vertices 1 and 2 are on the outer face of Gk (co3) vertex k+1 is on the outer face of Gk and the neighbor of k+1 is consecutive on the outer cycle Co(Gk).

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 (co1)-(co3) 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 ≠ v1, v2 on the cycle Co(Gk) which is not an end of a chord of Co(Gk) then clearly (co1)-(co3) hold for k-1 since Gk-1 = Gk – vk.

Now we have to proof that there is such a vertex. w = vk Gk-1 v2 v1 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) = w1, w2, …., wt, where w1= v1 and wt = v2.

In Case (i), any of the vertices w2, w3, …, wt-1 is such a vertex w. wq-1 wq wp+1 wp w2 w1=v1 wt=v2 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 13 14 C0(G14) 15 10 12 11 Co(G15) 9 6 C0(G13) 8 5 7 4 3 2 1 v2 v1 Choose a vertex x such that chords(x) = 0 and x ≠ v1, v2

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.

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

Straight Line Grid Drawing Plane graph de Fraysseix ‘90

Schnyder ‘90 H n-2 n-2 W

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

de Fraysseix ‘90 Shift Method Schnyder ‘90 Realizer Method

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Some other vertex orsering Antibandwidth labeling Graceful labeling

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