n u 1 u 2 u Canonical Decomposition

V 8 V 7 V 6 V 5 V 4 V 3 V 2 V 1 n u 1 u 2 u

V 8 V 7 V 6 V 5 V 4 V 3 V 2 V 1 n u 1 u 2 u (cd1) V 1 is the set of all vertices on the inner face containing edge (u 1,u 2 ) ． (cd2) for each index k, 1 ≤ k ≤ h, G k is internally 3-connected. (cd3) for each k, 2 ≦ k ≦ h-1, vertices in V k are on the outer vertices and the following (a) and (b) holds ． Canonical Decomposition

V 8 V 7 V 6 V 5 V 4 V 3 V 2 V 1 n u 1 u 2 u (cd1) V 1 is the set of all vertices on the inner face containing edge (u 1,u 2 ) ． (cd2) for each index k, 1 ≤ k ≤ h, G k is internally 3-connected. (cd3) for each k, 2 ≦ k ≦ h-1, vertices in V k are on the outer vertices and the following (a) and (b) holds ．

separation pair u v Internally 3-connected G is bi-connected For any separation pair {u,v} of G u and v are outer vertices each connected component of G-{u,v} contains an outer vertex.

Internally 3-connected G is bi-connected For any separation pair {u,v} of G u and v are outer vertices each connected component of G-{u,v} contains an outer vertex.

v 2 separation pair u v Internally 3-connected G is bi-connected For any separation pair {u,v} of G u and v are outer vertices each connected component of G-{u,v} contains an outer vertex.

Internally 3-connected G is bi-connected For any separation pair {u,v} of G u and v are outer vertices each connected component of G-{u,v} contains an outer vertex.

Internally 3-connected G is bi-connected For any separation pair {u,v} of G u and v are outer vertices each connected component of G-{u,v} contains an outer vertex.

Internally 3-connected G is bi-connected For any separation pair {u,v} of G u and v are outer vertices each connected component of G-{u,v} contains an outer vertex.

Internally 3-connected G is bi-connected For any separation pair {u,v} of G u and v are outer vertices each connected component of G-{u,v} contains an outer vertex.

Internally 3-connected G is bi-connected For any separation pair {u,v} of G u and v are outer vertices each connected component of G-{u,v} contains an outer vertex.

Internally 3-connected G is bi-connected For any separation pair {u,v} of G u and v are outer vertices each connected component of G-{u,v} contains an outer vertex.

Internally 3-connected G is bi-connected For any separation pair {u,v} of G u and v are outer vertices each connected component of G-{u,v} contains an outer vertex.

Internally 3-connected G is bi-connected For any separation pair {u,v} of G u and v are outer vertices each connected component of G-{u,v} contains an outer vertex.

Internally 3-connected G is bi-connected For any separation pair {u,v} of G u and v are outer vertices each connected component of G-{u,v} contains an outer vertex.

Internally 3-connected G is bi-connected For any separation pair {u,v} of G u and v are outer vertices each connected component of G-{u,v} contains an outer vertex.

Internally 3-connected G is bi-connected For any separation pair {u,v} of G u and v are outer vertices each connected component of G-{u,v} contains an outer vertex.

V 8 V 7 V 6 V 5 V 4 V 3 V 2 V 1 n u 1 u 2 u Canonical Decomposition (cd1) V 1 is the set of all vertices on the inner face containing edge (u 1,u 2 ) ． (cd2) for each index k, 1 ≤ k ≤ h, G k is internally 3-connected. (cd3) for each k, 2 ≦ k ≦ h-1, vertices in V k are on the outer vertices and the following (a) and (b) holds ．

V 1 V 8 V 7 V 6 V 5 V 4 V 3 V 2 n u 1 u 2 u G 1 Canonical Decomposition (cd1) V 1 is the set of all vertices on the inner face containing edge (u 1,u 2 ) ． (cd2) for each index k, 1 ≤ k ≤ h, G k is internally 3-connected. (cd3) for each k, 2 ≦ k ≦ h-1, vertices in V k are on the outer vertices and the following (a) and (b) holds ．

V 1 V 8 V 7 V 6 V 5 V 4 V 3 n u 1 u 2 u V 2 G 2 Canonical Decomposition (cd1) V 1 is the set of all vertices on the inner face containing edge (u 1,u 2 ) ． (cd2) for each index k, 1 ≤ k ≤ h, G k is internally 3-connected. (cd3) for each k, 2 ≦ k ≦ h-1, vertices in V k are on the outer vertices and the following (a) and (b) holds ．

V 1 V 8 V 7 V 6 V 5 V 4 V 3 n u 1 u 2 u V 2 G 3 Canonical Decomposition (cd1) V 1 is the set of all vertices on the inner face containing edge (u 1,u 2 ) ． (cd2) for each index k, 1 ≤ k ≤ h, G k is internally 3-connected. (cd3) for each k, 2 ≦ k ≦ h-1, vertices in V k are on the outer vertices and the following (a) and (b) holds ．

V 1 V 8 V 7 V 6 V 5 V 4 V 3 n u 1 u 2 u V 2 G 4 Canonical Decomposition (cd1) V 1 is the set of all vertices on the inner face containing edge (u 1,u 2 ) ． (cd2) for each index k, 1 ≤ k ≤ h, G k is internally 3-connected. (cd3) for each k, 2 ≦ k ≦ h-1, vertices in V k are on the outer vertices and the following (a) and (b) holds ．

V 1 V 8 V 7 V 6 V 5 V 4 V 3 n u 1 u 2 u V 2 G 5 Canonical Decomposition (cd1) V 1 is the set of all vertices on the inner face containing edge (u 1,u 2 ) ． (cd2) for each index k, 1 ≤ k ≤ h, G k is internally 3-connected. (cd3) for each k, 2 ≦ k ≦ h-1, vertices in V k are on the outer vertices and the following (a) and (b) holds ．

V 1 V 8 V 7 V 6 V 5 V 4 V 3 n u 1 u 2 u V 2 G 6 Canonical Decomposition (cd1) V 1 is the set of all vertices on the inner face containing edge (u 1,u 2 ) ． (cd2) for each index k, 1 ≤ k ≤ h, G k is internally 3-connected. (cd3) for each k, 2 ≦ k ≦ h-1, vertices in V k are on the outer vertices and the following (a) and (b) holds ．

V 1 V 8 V 7 V 6 V 5 V 4 V 3 n u 1 u 2 u V 2 G 7 Canonical Decomposition (cd1) V 1 is the set of all vertices on the inner face containing edge (u 1,u 2 ) ． (cd2) for each index k, 1 ≤ k ≤ h, G k is internally 3-connected. (cd3) for each k, 2 ≦ k ≦ h-1, vertices in V k are on the outer vertices and the following (a) and (b) holds ．

V 1 V 8 V 7 V 6 V 5 V 4 V 3 n u 1 u 2 u V 2 G 8 Canonical Decomposition (cd1) V 1 is the set of all vertices on the inner face containing edge (u 1,u 2 ) ． (cd2) for each index k, 1 ≤ k ≤ h, G k is internally 3-connected. (cd3) for each k, 2 ≦ k ≦ h-1, vertices in V k are on the outer vertices and the following (a) and (b) holds ．

V 1 V 8 V 7 V 6 V 5 V 4 V 3 n u 1 u 2 u V 2 G k-1 (a) G k-1 (b) VkVk VkVk Canonical Decomposition (cd1) V 1 is the set of all vertices on the inner face containing edge (u 1,u 2 ) ． (cd2) for each index k, 1 ≤ k ≤ h, G k is internally 3-connected. (cd3) for each k, 2 ≦ k ≦ h-1, vertices in V k are on the outer vertices and the following (a) and (b) holds ．

V 1 V 8 V 7 V 6 V 5 V 4 V 3 n u 1 u 2 u V 2 G k-1 (a) G k-1 (b) VkVk VkVk G 2 Canonical Decomposition (cd1) V 1 is the set of all vertices on the inner face containing edge (u 1,u 2 ) ． (cd2) for each index k, 1 ≤ k ≤ h, G k is internally 3-connected. (cd3) for each k, 2 ≦ k ≦ h-1, vertices in V k are on the outer vertices and the following (a) and (b) holds ．

V 1 V 8 V 7 V 6 V 5 V 4 V 3 n u 1 u 2 u V 2 G k-1 (a) G k-1 (b) VkVk VkVk G 6 Canonical Decomposition (cd1) V 1 is the set of all vertices on the inner face containing edge (u 1,u 2 ) ． (cd2) for each index k, 1 ≤ k ≤ h, G k is internally 3-connected. (cd3) for each k, 2 ≦ k ≦ h-1, vertices in V k are on the outer vertices and the following (a) and (b) holds ．

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

U3U3 G k G k -1 (a) G k G k -1 G k G (b) (c) face U k U k U k U 12 U 11 U 10 U9U9 U8U8 U7U7 U6U6 U5U5 U4U4 U2U2 4-canonical decomposition[NRN97] U1U1 (a generalization of st-numbering) G

G k G k -1 (a) G k G k -1 G k G (b) (c) face U 12 U 11 U 10 U9U9 U8U8 U7U7 U6U6 U5U5 U4U4 U3U3 U2U2 4-canonical decomposition[NRN97] U1U1 (a generalization of st-numbering)

U3U3 4-canonical decomposition[NRN97] (a generalization of st-numbering) G k -1 G k G k G (a) G k G k -1 G k G (b) (c) face

G k G k -1 (a) G k G k -1 G k G (b) (c) face U 12 U 11 U 10 U9U9 U8U8 U7U7 U6U6 U5U5 U4U4 U3U3 U2U2 4-canonical decomposition[NRN97] U1U1 (a generalization of st-numbering)

U9U9 4-canonical decomposition[NRN97] (a generalization of st-numbering) G k -1 G k face G k G k -1 (a) G k G k -1 G k G (b) (c) face

G k G k -1 (a) G k G k -1 G k G (b) (c) face U 12 U 11 U 10 U9U9 U8U8 U7U7 U6U6 U5U5 U4U4 U3U3 U2U2 4-canonical decomposition[NRN97] U1U1 (a generalization of st-numbering)

U5U5 4-canonical decomposition[NRN97] (a generalization of st-numbering) G k -1 G k face G k G k -1 (a) G k G k -1 G k G (b) (c) face

U1U1 U 12 U 11 U 10 U9U9 U8U8 U7U7 U6U6 U5U5 U4U4 U3U3 U2U2 4-canonical decomposition[NRN97] n = 21 (a generalization of st-numbering) G k G k -1 (a) G k G k -1 G k G (b) (c) face U k U k U k G O(n)O(n)

Conclusions st-numbering Canonical ordering 4-canonical ordering Canonical decomposition 4-canonical decomposition

Conclusions Recent Development Orderly spanning treesChiang et al., 2001 Miura et al Canonical decomposition, realizer and orderly Spanning tree are equivalent notions.