Descriptive Complexity of Graph Classes Via Labeling Schemes

Descriptive Complexity of Graph Classes via Labeling Schemes

Maurice Chandoo FernUniversität in Hagen

ACTO Seminar University of Liverpool May 2021 2 7 1 3 4 5 6 8 9 10

label `(v) ∈ {1,..., 2n} × {1,..., 2n} adjacency matrix 2(1+log n)  0 ··· 1  labeling `: V(G) → {0, 1} . . .  . .. .  1 ··· 0 requires n2 bits u `(u) 0 {u, v} ∈/ E(G) AIntv alternative representation v `(v) {u, v} ∈ E(G) {(1, 3), (2, 7), (4, 5),... } 1 requires n ·O(log n) bits label decoding algorithm

What is a Labeling Scheme?

interval graph G

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 2/16 2 7 1 3 4 5 6 8 9 10

What is a Labeling Scheme?

interval graph G

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 2/16 2 7 1 3 4 5 6 8 9 10

label `(v) ∈ {1,..., 2n} × {1,..., 2n} 2(1+log n)  0 ··· 1  labeling `: V(G) → {0, 1} . . .  . .. .  1 ··· 0 requires n2 bits u `(u) 0 {u, v} ∈/ E(G) AIntv alternative representation v `(v) {u, v} ∈ E(G) {(1, 3), (2, 7), (4, 5),... } 1 requires n ·O(log n) bits label decoding algorithm

What is a Labeling Scheme?

interval graph G

adjacency matrix

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 2/16 2 7 1 3 4 5 6 8 9 10

What is a Labeling Scheme?

2 3 4 5

interval graph G

adjacency matrix

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 2/16 2 7 1 3 4 5 6 8 9 10

label `(v) ∈ {1,..., 2n} × {1,..., 2n} labeling `: V(G) → {0, 1}2(1+log n)

requires n2 bits u `(u) 0 {u, v} ∈/ E(G) AIntv alternative representation v `(v) {u, v} ∈ E(G) {(1, 3), (2, 7), (4, 5),... } 1 requires n ·O(log n) bits label decoding algorithm

What is a Labeling Scheme?

2 3 4 5

interval graph G

adjacency matrix  0 ··· 1  . . .  . .. .  1 ··· 0

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 2/16 2 7 1 3 4 5 6 8 9 10

label `(v) ∈ {1,..., 2n} × {1,..., 2n} labeling `: V(G) → {0, 1}2(1+log n)

u `(u) 0 {u, v} ∈/ E(G) AIntv alternative representation v `(v) {u, v} ∈ E(G) {(1, 3), (2, 7), (4, 5),... } 1 requires n ·O(log n) bits label decoding algorithm

What is a Labeling Scheme?

2 3 4 5

interval graph G

adjacency matrix  0 ··· 1  . . .  . .. .  1 ··· 0 requires n2 bits

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 2/16 2 7 1 3 4 5 6 8 9 10

label `(v) ∈ {1,..., 2n} × {1,..., 2n} labeling `: V(G) → {0, 1}2(1+log n)

u `(u) 0 {u, v} ∈/ E(G) AIntv v `(v) {u, v} ∈ E(G) {(1, 3), (2, 7), (4, 5),... } 1 requires n ·O(log n) bits label decoding algorithm

What is a Labeling Scheme?

2 3 4 5

interval graph G

adjacency matrix  0 ··· 1  . . .  . .. .  1 ··· 0 requires n2 bits

alternative representation

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 2/16 2 7 1 3 4 5 6 8 9 10

label `(v) ∈ {1,..., 2n} × {1,..., 2n} labeling `: V(G) → {0, 1}2(1+log n)

u `(u) 0 {u, v} ∈/ E(G) AIntv v `(v) {u, v} ∈ E(G) {(1, 3), (2, 7), (4, 5),... } 1 requires n ·O(log n) bits label decoding algorithm

What is a Labeling Scheme?

interval graph G

adjacency matrix  0 ··· 1  . . .  . .. .  1 ··· 0 requires n2 bits

alternative representation

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 2/16 2 7 3 4 5 6 8 9 10

label `(v) ∈ {1,..., 2n} × {1,..., 2n} labeling `: V(G) → {0, 1}2(1+log n)

u `(u) 0 {u, v} ∈/ E(G) AIntv v `(v) {u, v} ∈ E(G) {(1, 3), (2, 7), (4, 5),... } 1 requires n ·O(log n) bits label decoding algorithm

What is a Labeling Scheme?

interval graph G

adjacency matrix  0 ··· 1  . . .  . .. .  1 ··· 0 requires n2 bits

alternative representation

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 2/16 7 3 4 5 6 8 9 10

label `(v) ∈ {1,..., 2n} × {1,..., 2n} labeling `: V(G) → {0, 1}2(1+log n)

u `(u) 0 {u, v} ∈/ E(G) AIntv v `(v) {u, v} ∈ E(G) {(1, 3), (2, 7), (4, 5),... } 1 requires n ·O(log n) bits label decoding algorithm

What is a Labeling Scheme?

2 1

interval graph G

adjacency matrix  0 ··· 1  . . .  . .. .  1 ··· 0 requires n2 bits

alternative representation

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 2/16 7 4 5 6 8 9 10

label `(v) ∈ {1,..., 2n} × {1,..., 2n} labeling `: V(G) → {0, 1}2(1+log n)

u `(u) 0 {u, v} ∈/ E(G) AIntv v `(v) {u, v} ∈ E(G) {(1, 3), (2, 7), (4, 5),... } 1 requires n ·O(log n) bits label decoding algorithm

What is a Labeling Scheme?

2 1 3

interval graph G

adjacency matrix  0 ··· 1  . . .  . .. .  1 ··· 0 requires n2 bits

alternative representation

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 2/16 7 5 6 8 9 10

label `(v) ∈ {1,..., 2n} × {1,..., 2n} labeling `: V(G) → {0, 1}2(1+log n)

u `(u) 0 {u, v} ∈/ E(G) AIntv v `(v) {u, v} ∈ E(G) {(1, 3), (2, 7), (4, 5),... } 1 requires n ·O(log n) bits label decoding algorithm

What is a Labeling Scheme?

2 1 3 4

interval graph G

adjacency matrix  0 ··· 1  . . .  . .. .  1 ··· 0 requires n2 bits

alternative representation

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 2/16 7 6 8 9 10

label `(v) ∈ {1,..., 2n} × {1,..., 2n} labeling `: V(G) → {0, 1}2(1+log n)

u `(u) 0 {u, v} ∈/ E(G) AIntv v `(v) {u, v} ∈ E(G) {(1, 3), (2, 7), (4, 5),... } 1 requires n ·O(log n) bits label decoding algorithm

What is a Labeling Scheme?

2 1 3 4 5

interval graph G

adjacency matrix  0 ··· 1  . . .  . .. .  1 ··· 0 requires n2 bits

alternative representation

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 2/16 7 8 9 10

label `(v) ∈ {1,..., 2n} × {1,..., 2n} labeling `: V(G) → {0, 1}2(1+log n)

u `(u) 0 {u, v} ∈/ E(G) AIntv v `(v) {u, v} ∈ E(G) {(1, 3), (2, 7), (4, 5),... } 1 requires n ·O(log n) bits label decoding algorithm

What is a Labeling Scheme?

2 1 3 4 5 6

interval graph G

adjacency matrix  0 ··· 1  . . .  . .. .  1 ··· 0 requires n2 bits

alternative representation

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 2/16 8 9 10

label `(v) ∈ {1,..., 2n} × {1,..., 2n} labeling `: V(G) → {0, 1}2(1+log n)

u `(u) 0 {u, v} ∈/ E(G) AIntv v `(v) {u, v} ∈ E(G) {(1, 3), (2, 7), (4, 5),... } 1 requires n ·O(log n) bits label decoding algorithm

What is a Labeling Scheme?

2 7 1 3 4 5 6

interval graph G

adjacency matrix  0 ··· 1  . . .  . .. .  1 ··· 0 requires n2 bits

alternative representation

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 2/16 9 10

label `(v) ∈ {1,..., 2n} × {1,..., 2n} labeling `: V(G) → {0, 1}2(1+log n)

u `(u) 0 {u, v} ∈/ E(G) AIntv v `(v) {u, v} ∈ E(G) {(1, 3), (2, 7), (4, 5),... } 1 requires n ·O(log n) bits label decoding algorithm

What is a Labeling Scheme?

2 7 1 3 4 5 6 8

interval graph G

adjacency matrix  0 ··· 1  . . .  . .. .  1 ··· 0 requires n2 bits

alternative representation

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 2/16 10

label `(v) ∈ {1,..., 2n} × {1,..., 2n} labeling `: V(G) → {0, 1}2(1+log n)

u `(u) 0 {u, v} ∈/ E(G) AIntv v `(v) {u, v} ∈ E(G) {(1, 3), (2, 7), (4, 5),... } 1 requires n ·O(log n) bits label decoding algorithm

What is a Labeling Scheme?

2 7 1 3 4 5 6 8 9

interval graph G

adjacency matrix  0 ··· 1  . . .  . .. .  1 ··· 0 requires n2 bits

alternative representation

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 2/16 label `(v) ∈ {1,..., 2n} × {1,..., 2n} labeling `: V(G) → {0, 1}2(1+log n)

u `(u) 0 {u, v} ∈/ E(G) AIntv v `(v) {u, v} ∈ E(G) {(1, 3), (2, 7), (4, 5),... } 1 requires n ·O(log n) bits label decoding algorithm

What is a Labeling Scheme?

2 7 1 3 4 5 6 8 9 10

interval graph G

adjacency matrix  0 ··· 1  . . .  . .. .  1 ··· 0 requires n2 bits

alternative representation

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 2/16 label `(v) ∈ {1,..., 2n} × {1,..., 2n} labeling `: V(G) → {0, 1}2(1+log n)

u `(u) 0 {u, v} ∈/ E(G) AIntv v `(v) {u, v} ∈ E(G) {(1, 3), (2, 7), (4, 5),... } 1 requires n ·O(log n) bits label decoding algorithm

What is a Labeling Scheme?

2 7 1, 3 1 3 4 5 6 8 9 10

interval graph G

adjacency matrix  0 ··· 1  . . .  . .. .  1 ··· 0 requires n2 bits

alternative representation

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 2/16 label `(v) ∈ {1,..., 2n} × {1,..., 2n} labeling `: V(G) → {0, 1}2(1+log n)

u `(u) 0 {u, v} ∈/ E(G) AIntv v `(v) {u, v} ∈ E(G) {(1, 3), (2, 7), (4, 5),... } 1 requires n ·O(log n) bits label decoding algorithm

What is a Labeling Scheme?

2, 7 2 7 1, 3 1 3 4 5 6 8 9 10

interval graph G

adjacency matrix  0 ··· 1  . . .  . .. .  1 ··· 0 requires n2 bits

alternative representation

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 2/16 label `(v) ∈ {1,..., 2n} × {1,..., 2n} labeling `: V(G) → {0, 1}2(1+log n)

u `(u) 0 {u, v} ∈/ E(G) AIntv v `(v) {u, v} ∈ E(G) {(1, 3), (2, 7), (4, 5),... } 1 requires n ·O(log n) bits label decoding algorithm

What is a Labeling Scheme?

2, 7 2 7 1, 3 4, 5 1 3 4 5 6 8 9 10

interval graph G

adjacency matrix  0 ··· 1  . . .  . .. .  1 ··· 0 requires n2 bits

alternative representation

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 2/16 label `(v) ∈ {1,..., 2n} × {1,..., 2n} labeling `: V(G) → {0, 1}2(1+log n)

u `(u) 0 {u, v} ∈/ E(G) AIntv v `(v) {u, v} ∈ E(G) {(1, 3), (2, 7), (4, 5),... } 1 requires n ·O(log n) bits label decoding algorithm

What is a Labeling Scheme?

2, 7 2 7 1, 3 4, 5 6, 8 1 3 4 5 6 8 9 10

interval graph G

adjacency matrix  0 ··· 1  . . .  . .. .  1 ··· 0 requires n2 bits

alternative representation

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 2/16 label `(v) ∈ {1,..., 2n} × {1,..., 2n} labeling `: V(G) → {0, 1}2(1+log n)

u `(u) 0 {u, v} ∈/ E(G) AIntv v `(v) {u, v} ∈ E(G) {(1, 3), (2, 7), (4, 5),... } 1 requires n ·O(log n) bits label decoding algorithm

What is a Labeling Scheme?

2, 7 2 7 1, 3 4, 5 6, 8 9, 10 1 3 4 5 6 8 9 10

interval graph G

adjacency matrix  0 ··· 1  . . .  . .. .  1 ··· 0 requires n2 bits

alternative representation

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 2/16 label `(v) ∈ {1,..., 2n} × {1,..., 2n} labeling `: V(G) → {0, 1}2(1+log n)

u `(u) 0 {u, v} ∈/ E(G) AIntv v `(v) 1 {u, v} ∈ E(G)

requires n ·O(log n) bits label decoding algorithm

What is a Labeling Scheme?

2, 7 2 7 1, 3 4, 5 6, 8 9, 10 1 3 4 5 6 8 9 10

interval graph G

adjacency matrix  0 ··· 1  . . .  . .. .  1 ··· 0 requires n2 bits

alternative representation {(1, 3), (2, 7), (4, 5),... }

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 2/16 labeling `: V(G) → {0, 1}2(1+log n)

u `(u) 0 {u, v} ∈/ E(G) AIntv v `(v) 1 {u, v} ∈ E(G)

requires n ·O(log n) bits label decoding algorithm

What is a Labeling Scheme?

2, 7 2 7 1, 3 4, 5 6, 8 9, 10 1 3 4 5 6 8 9 10

interval graph G

label `(v) ∈ {1,..., 2n} × {1,..., 2n} adjacency matrix  0 ··· 1  . . .  . .. .  1 ··· 0 requires n2 bits

alternative representation {(1, 3), (2, 7), (4, 5),... }

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 2/16 labeling `: V(G) → {0, 1}2(1+log n)

u `(u) 0 {u, v} ∈/ E(G) AIntv v `(v) 1 {u, v} ∈ E(G)

requires n ·O(log n) bits label decoding algorithm

What is a Labeling Scheme?

2, 7 2 7 1, 3 4, 5 6, 8 9, 10 1 3 4 5 6 8 9 10

interval graph G

label `(v) ∈ {1,..., 2n} × {1,..., 2n} adjacency matrix  0 ··· 1  . . .  . .. .  1 ··· 0 requires n2 bits

alternative representation {(1, 3), (2, 7), (4, 5),... }

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 2/16 labeling `: V(G) → {0, 1}2(1+log n)

u `(u) 0 {u, v} ∈/ E(G) AIntv v `(v) 1 {u, v} ∈ E(G)

label decoding algorithm

What is a Labeling Scheme?

2, 7 2 7 1, 3 4, 5 6, 8 9, 10 1 3 4 5 6 8 9 10

interval graph G

label `(v) ∈ {1,..., 2n} × {1,..., 2n} adjacency matrix  0 ··· 1  . . .  . .. .  1 ··· 0 requires n2 bits

alternative representation {(1, 3), (2, 7), (4, 5),... } requires n ·O(log n) bits

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 2/16 labeling `: V(G) → {0, 1}2(1+log n)

u `(u) 0 {u, v} ∈/ E(G) AIntv v `(v) 1 {u, v} ∈ E(G)

label decoding algorithm

What is a Labeling Scheme?

2, 7 2 7 1, 3 4, 5 6, 8 9, 10 1 3 4 5 6 8 9 10

interval graph G

label `(v) ∈ {1,..., 2n} × {1,..., 2n} adjacency matrix  0 ··· 1  . . .  . .. .  1 ··· 0 requires n2 bits

alternative representation {(1, 3), (2, 7), (4, 5),... } requires n ·O(log n) bits

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 2/16 u `(u) 0 {u, v} ∈/ E(G) AIntv v `(v) 1 {u, v} ∈ E(G)

label decoding algorithm

What is a Labeling Scheme?

2, 7 2 7 1, 3 4, 5 6, 8 9, 10 1 3 4 5 6 8 9 10

interval graph G

label `(v) ∈ {1,..., 2n} × {1,..., 2n} adjacency matrix 2(1+log n)  0 ··· 1  labeling `: V(G) → {0, 1} . . .  . .. .  1 ··· 0 requires n2 bits

alternative representation {(1, 3), (2, 7), (4, 5),... } requires n ·O(log n) bits

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 2/16 `(u) 0 {u, v} ∈/ E(G) AIntv `(v) 1 {u, v} ∈ E(G)

label decoding algorithm

What is a Labeling Scheme?

2, 7 2 7 1, 3 4, 5 6, 8 9, 10 1 3 4 5 6 8 9 10

interval graph G

label `(v) ∈ {1,..., 2n} × {1,..., 2n} adjacency matrix 2(1+log n)  0 ··· 1  labeling `: V(G) → {0, 1} . . .  . .. .  1 ··· 0 2 requires n bits u

alternative representation v {(1, 3), (2, 7), (4, 5),... } requires n ·O(log n) bits

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 2/16 0 {u, v} ∈/ E(G) AIntv 1 {u, v} ∈ E(G)

label decoding algorithm

What is a Labeling Scheme?

2, 7 2 7 1, 3 4, 5 6, 8 9, 10 1 3 4 5 6 8 9 10

interval graph G

label `(v) ∈ {1,..., 2n} × {1,..., 2n} adjacency matrix 2(1+log n)  0 ··· 1  labeling `: V(G) → {0, 1} . . .  . .. .  1 ··· 0 2 requires n bits u `(u)

alternative representation v `(v) {(1, 3), (2, 7), (4, 5),... } requires n ·O(log n) bits

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 2/16 0 {u, v} ∈/ E(G)

1 {u, v} ∈ E(G)

label decoding algorithm

What is a Labeling Scheme?

2, 7 2 7 1, 3 4, 5 6, 8 9, 10 1 3 4 5 6 8 9 10

interval graph G

label `(v) ∈ {1,..., 2n} × {1,..., 2n} adjacency matrix 2(1+log n)  0 ··· 1  labeling `: V(G) → {0, 1} . . .  . .. .  1 ··· 0 2 requires n bits u `(u) AIntv alternative representation v `(v) {(1, 3), (2, 7), (4, 5),... } requires n ·O(log n) bits

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 2/16 0 {u, v} ∈/ E(G)

1 {u, v} ∈ E(G)

label decoding algorithm

What is a Labeling Scheme?

2, 7 2 7 1, 3 4, 5 6, 8 9, 10 1 3 4 5 6 8 9 10

interval graph G

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 2/16 0 {u, v} ∈/ E(G)

1 {u, v} ∈ E(G)

label decoding algorithm

What is a Labeling Scheme?

2, 7 2 7 1, 3 4, 5 6, 8 9, 10 1 3 4 5 6 8 9 10

interval graph G

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 2/16 {u, v} ∈/ E(G)

1 {u, v} ∈ E(G)

label decoding algorithm

What is a Labeling Scheme?

2, 7 2 7 1, 3 4, 5 6, 8 9, 10 1 3 4 5 6 8 9 10

interval graph G

label `(v) ∈ {1,..., 2n} × {1,..., 2n} adjacency matrix 2(1+log n)  0 ··· 1  labeling `: V(G) → {0, 1} . . .  . .. .  1 ··· 0 requires n2 bits u `(u) 0 AIntv alternative representation v `(v) {(1, 3), (2, 7), (4, 5),... } requires n ·O(log n) bits

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 2/16 1 {u, v} ∈ E(G)

label decoding algorithm

What is a Labeling Scheme?

2, 7 2 7 1, 3 4, 5 6, 8 9, 10 1 3 4 5 6 8 9 10

interval graph G

label `(v) ∈ {1,..., 2n} × {1,..., 2n} adjacency matrix 2(1+log n)  0 ··· 1  labeling `: V(G) → {0, 1} . . .  . .. .  1 ··· 0 requires n2 bits u `(u) 0 {u, v} ∈/ E(G) AIntv alternative representation v `(v) {(1, 3), (2, 7), (4, 5),... } requires n ·O(log n) bits

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 2/16 {u, v} ∈ E(G)

label decoding algorithm

What is a Labeling Scheme?

2, 7 2 7 1, 3 4, 5 6, 8 9, 10 1 3 4 5 6 8 9 10

interval graph G

label `(v) ∈ {1,..., 2n} × {1,..., 2n} adjacency matrix 2(1+log n)  0 ··· 1  labeling `: V(G) → {0, 1} . . .  . .. .  1 ··· 0 requires n2 bits u `(u) 0 {u, v} ∈/ E(G) AIntv alternative representation v `(v) {(1, 3), (2, 7), (4, 5),... } 1 requires n ·O(log n) bits

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 2/16 label decoding algorithm

What is a Labeling Scheme?

2, 7 2 7 1, 3 4, 5 6, 8 9, 10 1 3 4 5 6 8 9 10

interval graph G

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 2/16 What is a Labeling Scheme?

2, 7 2 7 1, 3 4, 5 6, 8 9, 10 1 3 4 5 6 8 9 10

interval graph G

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 2/16 id(u) ∈ nbr(v) or id(v) ∈ nbr(u)

id(v1),..., id(v5) | {z } nbr(u)

I assign vertices of planar graph G short labels s.t. adjacency can be determined from them I assign each vertex of G an id {1,..., n} I every planar graph has a vertex with degree at most 5 I {u, v} ∈ E(G) ⇐⇒ I label length6 log n bits

v2 • pick a vertex u with deg(u) ≤ 5 v3 • set `(u) := id(u), u v1

v4 v5 • delete u and repeat

Labeling Scheme for Planar Graphs

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 3/16 id(u) ∈ nbr(v) or id(v) ∈ nbr(u)

id(v1),..., id(v5) | {z } nbr(u)

I assign each vertex of G an id {1,..., n} I every planar graph has a vertex with degree at most 5 I {u, v} ∈ E(G) ⇐⇒ I label length6 log n bits

v2 • pick a vertex u with deg(u) ≤ 5 v3 • set `(u) := id(u), u v1

v4 v5 • delete u and repeat

Labeling Scheme for Planar Graphs

I assign vertices of planar graph G short labels s.t. adjacency can be determined from them

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 3/16 id(u) ∈ nbr(v) or id(v) ∈ nbr(u)

id(v1),..., id(v5) | {z } nbr(u)

I every planar graph has a vertex with degree at most 5 I {u, v} ∈ E(G) ⇐⇒ I label length6 log n bits

v2 • pick a vertex u with deg(u) ≤ 5 v3 • set `(u) := id(u), u v1

v4 v5 • delete u and repeat

Labeling Scheme for Planar Graphs

I assign vertices of planar graph G short labels s.t. adjacency can be determined from them I assign each vertex of G an id {1,..., n}

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 3/16 id(u) ∈ nbr(v) or id(v) ∈ nbr(u)

id(v1),..., id(v5) | {z } nbr(u)

I {u, v} ∈ E(G) ⇐⇒ I label length6 log n bits

v2 • pick a vertex u with deg(u) ≤ 5 v3 • set `(u) := id(u), u v1

v4 v5 • delete u and repeat

Labeling Scheme for Planar Graphs

I assign vertices of planar graph G short labels s.t. adjacency can be determined from them I assign each vertex of G an id {1,..., n} I every planar graph has a vertex with degree at most 5

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 3/16 id(u) ∈ nbr(v) or id(v) ∈ nbr(u)

id(v1),..., id(v5) | {z } nbr(u)

I {u, v} ∈ E(G) ⇐⇒ I label length6 log n bits

v2 v3 • set `(u) := id(u), u v1

v4 v5 • delete u and repeat

Labeling Scheme for Planar Graphs

I assign vertices of planar graph G short labels s.t. adjacency can be determined from them I assign each vertex of G an id {1,..., n} I every planar graph has a vertex with degree at most 5

• pick a vertex u with deg(u) ≤ 5

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 3/16 id(u) ∈ nbr(v) or id(v) ∈ nbr(u)

id(v1),..., id(v5) | {z } nbr(u)

I {u, v} ∈ E(G) ⇐⇒ I label length6 log n bits

• set `(u) := id(u),

• delete u and repeat

Labeling Scheme for Planar Graphs

I assign vertices of planar graph G short labels s.t. adjacency can be determined from them I assign each vertex of G an id {1,..., n} I every planar graph has a vertex with degree at most 5

v2 • pick a vertex u with deg(u) ≤ 5 v3 a

u v1

v4 a v5

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 3/16 id(u) ∈ nbr(v) or id(v) ∈ nbr(u)

I {u, v} ∈ E(G) ⇐⇒ I label length6 log n bits

| {z } nbr(u) • delete u and repeat

Labeling Scheme for Planar Graphs

I assign vertices of planar graph G short labels s.t. adjacency can be determined from them I assign each vertex of G an id {1,..., n} I every planar graph has a vertex with degree at most 5

v2 • pick a vertex u with deg(u) ≤ 5 v3 a • set `(u) := id(u), id(v1),..., id(v5) u v1

v4 a v5

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 3/16 id(u) ∈ nbr(v) or id(v) ∈ nbr(u)

I {u, v} ∈ E(G) ⇐⇒ I label length6 log n bits

• delete u and repeat

Labeling Scheme for Planar Graphs

I assign vertices of planar graph G short labels s.t. adjacency can be determined from them I assign each vertex of G an id {1,..., n} I every planar graph has a vertex with degree at most 5

v2 • pick a vertex u with deg(u) ≤ 5 v3 a v • set `(u) := id(u), id(v1),..., id(v5) u 1 | {z } nbr(u) v4 a v5

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 3/16 id(u) ∈ nbr(v) or id(v) ∈ nbr(u)

I {u, v} ∈ E(G) ⇐⇒ I label length6 log n bits

Labeling Scheme for Planar Graphs

I assign vertices of planar graph G short labels s.t. adjacency can be determined from them I assign each vertex of G an id {1,..., n} I every planar graph has a vertex with degree at most 5

v2 • pick a vertex u with deg(u) ≤ 5 v3 a v • set `(u) := id(u), id(v1),..., id(v5) u 1 | {z } nbr(u) v4 a v5 • delete u and repeat

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 3/16 id(u) ∈ nbr(v) or id(v) ∈ nbr(u) I label length6 log n bits

Labeling Scheme for Planar Graphs

v2 • pick a vertex u with deg(u) ≤ 5 v3 a v • set `(u) := id(u), id(v1),..., id(v5) u 1 | {z } nbr(u) v4 a v5 • delete u and repeat

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 3/16 or id(v) ∈ nbr(u) I label length6 log n bits

Labeling Scheme for Planar Graphs

v2 • pick a vertex u with deg(u) ≤ 5 v3 a v • set `(u) := id(u), id(v1),..., id(v5) u 1 | {z } nbr(u) v4 a v5 • delete u and repeat

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 3/16 I label length6 log n bits

Labeling Scheme for Planar Graphs

v2 • pick a vertex u with deg(u) ≤ 5 v3 a v • set `(u) := id(u), id(v1),..., id(v5) u 1 | {z } nbr(u) v4 a v5 • delete u and repeat

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 3/16 Labeling Scheme for Planar Graphs

v2 • pick a vertex u with deg(u) ≤ 5 v3 a v • set `(u) := id(u), id(v1),..., id(v5) u 1 | {z } nbr(u) v4 a v5 • delete u and repeat

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 3/16 O(n) O(n log n) |Cn| ∈ n = 2

I a label decoder F ⊆ {0, 1}∗ × {0, 1}∗ I a label length c ∈ N such that for every graph G ∈C there exists a labeling `: V(G) → {0, 1}c log n with

(u, v) ∈ E(G) ⇔ (`(u), `(v)) ∈ F

I space-eﬃcient representation for C with factorial speed

I introduced by Muller ’88 and Kannan et al ’92 I extensions: distance labeling schemes, variable label length, . . .

Adjacency Labeling Schemes

Deﬁnition A graph class C has a labeling scheme if there exist:

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 4/16 O(n) O(n log n) |Cn| ∈ n = 2

I a label length c ∈ N such that for every graph G ∈C there exists a labeling `: V(G) → {0, 1}c log n with

(u, v) ∈ E(G) ⇔ (`(u), `(v)) ∈ F

I space-eﬃcient representation for C with factorial speed

I introduced by Muller ’88 and Kannan et al ’92 I extensions: distance labeling schemes, variable label length, . . .

Adjacency Labeling Schemes

Deﬁnition A graph class C has a labeling scheme if there exist: I a label decoder F ⊆ {0, 1}∗ × {0, 1}∗

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 4/16 O(n) O(n log n) |Cn| ∈ n = 2

such that for every graph G ∈C there exists a labeling `: V(G) → {0, 1}c log n with

(u, v) ∈ E(G) ⇔ (`(u), `(v)) ∈ F

I space-eﬃcient representation for C with factorial speed

I introduced by Muller ’88 and Kannan et al ’92 I extensions: distance labeling schemes, variable label length, . . .

Adjacency Labeling Schemes

Deﬁnition A graph class C has a labeling scheme if there exist: I a label decoder F ⊆ {0, 1}∗ × {0, 1}∗ I a label length c ∈ N

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 4/16 O(n) O(n log n) |Cn| ∈ n = 2

(u, v) ∈ E(G) ⇔ (`(u), `(v)) ∈ F

I space-eﬃcient representation for C with factorial speed

I introduced by Muller ’88 and Kannan et al ’92 I extensions: distance labeling schemes, variable label length, . . .

Adjacency Labeling Schemes

Deﬁnition A graph class C has a labeling scheme if there exist: I a label decoder F ⊆ {0, 1}∗ × {0, 1}∗ I a label length c ∈ N such that for every graph G ∈C there exists a labeling `: V(G) → {0, 1}c log n with

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 4/16 O(n) O(n log n) |Cn| ∈ n = 2

I space-eﬃcient representation for C with factorial speed

I introduced by Muller ’88 and Kannan et al ’92 I extensions: distance labeling schemes, variable label length, . . .

Adjacency Labeling Schemes

(u, v) ∈ E(G) ⇔ (`(u), `(v)) ∈ F

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 4/16 O(n) O(n log n) |Cn| ∈ n = 2 I introduced by Muller ’88 and Kannan et al ’92 I extensions: distance labeling schemes, variable label length, . . .

Adjacency Labeling Schemes

(u, v) ∈ E(G) ⇔ (`(u), `(v)) ∈ F

I space-eﬃcient representation for C with factorial speed

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 4/16 = 2O(n log n) I introduced by Muller ’88 and Kannan et al ’92 I extensions: distance labeling schemes, variable label length, . . .

Adjacency Labeling Schemes

(u, v) ∈ E(G) ⇔ (`(u), `(v)) ∈ F

I space-eﬃcient representation for C with factorial speed O(n) |Cn| ∈ n

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 4/16 I introduced by Muller ’88 and Kannan et al ’92 I extensions: distance labeling schemes, variable label length, . . .

Adjacency Labeling Schemes

(u, v) ∈ E(G) ⇔ (`(u), `(v)) ∈ F

I space-eﬃcient representation for C with factorial speed O(n) O(n log n) |Cn| ∈ n = 2

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 4/16 I extensions: distance labeling schemes, variable label length, . . .

Adjacency Labeling Schemes

(u, v) ∈ E(G) ⇔ (`(u), `(v)) ∈ F

I space-eﬃcient representation for C with factorial speed O(n) O(n log n) |Cn| ∈ n = 2 I introduced by Muller ’88 and Kannan et al ’92

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 4/16 Adjacency Labeling Schemes

(u, v) ∈ E(G) ⇔ (`(u), `(v)) ∈ F

I space-eﬃcient representation for C with factorial speed O(n) O(n log n) |Cn| ∈ n = 2 I introduced by Muller ’88 and Kannan et al ’92 I extensions: distance labeling schemes, variable label length, . . .

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 4/16 6 log n {0, 1} Fplanar 6 |V(Gn)| = n

I Gn contains every planar graph on n vertices for all n I |V(Gn)| is polynomially bounded I labeling scheme for planar graphs: Gn = ( , )

I labeling scheme ⇒ polynomial universal graph I label length = degree of polynomial bounding |V(Gn)|

Polynomial Universal Graphs

I ﬁnd a family of graphs (Gn)n∈N such that

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 5/16 6 log n {0, 1} Fplanar 6 |V(Gn)| = n

I |V(Gn)| is polynomially bounded I labeling scheme for planar graphs: Gn = ( , )

I labeling scheme ⇒ polynomial universal graph I label length = degree of polynomial bounding |V(Gn)|

Polynomial Universal Graphs

I ﬁnd a family of graphs (Gn)n∈N such that I Gn contains every planar graph on n vertices for all n

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 5/16 6 log n {0, 1} Fplanar 6 |V(Gn)| = n

I labeling scheme for planar graphs: Gn = ( , )

I labeling scheme ⇒ polynomial universal graph I label length = degree of polynomial bounding |V(Gn)|

Polynomial Universal Graphs

I ﬁnd a family of graphs (Gn)n∈N such that I Gn contains every planar graph on n vertices for all n I |V(Gn)| is polynomially bounded

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 5/16 6 log n {0, 1} Fplanar 6 |V(Gn)| = n I labeling scheme ⇒ polynomial universal graph I label length = degree of polynomial bounding |V(Gn)|

Polynomial Universal Graphs

I ﬁnd a family of graphs (Gn)n∈N such that I Gn contains every planar graph on n vertices for all n I |V(Gn)| is polynomially bounded I labeling scheme for planar graphs: Gn = ( , )

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 5/16 Fplanar 6 |V(Gn)| = n I labeling scheme ⇒ polynomial universal graph I label length = degree of polynomial bounding |V(Gn)|

Polynomial Universal Graphs

I ﬁnd a family of graphs (Gn)n∈N such that I Gn contains every planar graph on n vertices for all n I |V(Gn)| is polynomially bounded I labeling scheme for planar graphs: 6 log n Gn = ({0, 1} , )

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 5/16 6 |V(Gn)| = n I labeling scheme ⇒ polynomial universal graph I label length = degree of polynomial bounding |V(Gn)|

Polynomial Universal Graphs

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 5/16 I labeling scheme ⇒ polynomial universal graph I label length = degree of polynomial bounding |V(Gn)|

Polynomial Universal Graphs

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 5/16 I label length = degree of polynomial bounding |V(Gn)|

Polynomial Universal Graphs

I ﬁnd a family of graphs (Gn)n∈N such that I Gn contains every planar graph on n vertices for all n I |V(Gn)| is polynomially bounded I labeling scheme for planar graphs: 6 log n Gn = ({0, 1} , Fplanar) 6 |V(Gn)| = n I labeling scheme ⇒ polynomial universal graph

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 5/16 I label length = degree of polynomial bounding |V(Gn)|

Polynomial Universal Graphs

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 5/16 Polynomial Universal Graphs

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 5/16 representations for intersection graph classes Köbler et at ’11, McDiarmid & Müller ’13 time to query an edge

minimally sized universal graphs Alstrup et al ’17, Atminas et al ’15 I encoding time (ﬁnding a labeling `):

I decoding time: what graph classes have a labeling scheme when label decoder F has low complexity?

Complexity Aspects of Labeling Schemes

I label length:

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 6/16 representations for intersection graph classes Köbler et at ’11, McDiarmid & Müller ’13 time to query an edge

Alstrup et al ’17, Atminas et al ’15 I encoding time (ﬁnding a labeling `):

I decoding time: what graph classes have a labeling scheme when label decoder F has low complexity?

Complexity Aspects of Labeling Schemes

I label length: minimally sized universal graphs

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 6/16 representations for intersection graph classes Köbler et at ’11, McDiarmid & Müller ’13 time to query an edge

I encoding time (ﬁnding a labeling `):

I decoding time: what graph classes have a labeling scheme when label decoder F has low complexity?

Complexity Aspects of Labeling Schemes

I label length: minimally sized universal graphs Alstrup et al ’17, Atminas et al ’15

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 6/16 time to query an edge

representations for intersection graph classes Köbler et at ’11, McDiarmid & Müller ’13 I decoding time: what graph classes have a labeling scheme when label decoder F has low complexity?

Complexity Aspects of Labeling Schemes

I label length: minimally sized universal graphs Alstrup et al ’17, Atminas et al ’15 I encoding time (ﬁnding a labeling `):

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 6/16 time to query an edge

Köbler et at ’11, McDiarmid & Müller ’13 I decoding time: what graph classes have a labeling scheme when label decoder F has low complexity?

Complexity Aspects of Labeling Schemes

I label length: minimally sized universal graphs Alstrup et al ’17, Atminas et al ’15 I encoding time (ﬁnding a labeling `): representations for intersection graph classes

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 6/16 time to query an edge

I decoding time: what graph classes have a labeling scheme when label decoder F has low complexity?

Complexity Aspects of Labeling Schemes

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 6/16 time to query an edge what graph classes have a labeling scheme when label decoder F has low complexity?

Complexity Aspects of Labeling Schemes

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 6/16 what graph classes have a labeling scheme when label decoder F has low complexity?

Complexity Aspects of Labeling Schemes

I label length: minimally sized universal graphs Alstrup et al ’17, Atminas et al ’15 I encoding time (ﬁnding a labeling `): representations for intersection graph classes Köbler et at ’11, McDiarmid & Müller ’13 I decoding time: time to query an edge

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 6/16 Complexity Aspects of Labeling Schemes

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 6/16 (x, y) ∈ FL ⇔ xy ∈ L

C ∈ GP ⇔ C has a labeling scheme (FL, c) with L ∈ P and c ∈ N

I a language L ⊆ {0, 1}∗ induces label decoder

I complexity class P graph class complexity class GP

I C ∈ GP implies polylog(n) query time

Complexity of Label Decoders

I label decoder F ⊆ {0, 1}∗ × {0, 1}∗

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 7/16 C ∈ GP ⇔ C has a labeling scheme (FL, c) with L ∈ P and c ∈ N

(x, y) ∈ FL ⇔ xy ∈ L

I complexity class P graph class complexity class GP

I C ∈ GP implies polylog(n) query time

Complexity of Label Decoders

I label decoder F ⊆ {0, 1}∗ × {0, 1}∗ I a language L ⊆ {0, 1}∗ induces label decoder

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 7/16 C ∈ GP ⇔ C has a labeling scheme (FL, c) with L ∈ P and c ∈ N

I complexity class P graph class complexity class GP

I C ∈ GP implies polylog(n) query time

Complexity of Label Decoders

I label decoder F ⊆ {0, 1}∗ × {0, 1}∗ I a language L ⊆ {0, 1}∗ induces label decoder

(x, y) ∈ FL ⇔ xy ∈ L

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 7/16 C ∈ GP ⇔ C has a labeling scheme (FL, c) with L ∈ P and c ∈ N

I C ∈ GP implies polylog(n) query time

Complexity of Label Decoders

I label decoder F ⊆ {0, 1}∗ × {0, 1}∗ I a language L ⊆ {0, 1}∗ induces label decoder

(x, y) ∈ FL ⇔ xy ∈ L

I complexity class P graph class complexity class GP

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 7/16 C has a labeling scheme (FL, c) with L ∈ P and c ∈ N

I C ∈ GP implies polylog(n) query time

Complexity of Label Decoders

I label decoder F ⊆ {0, 1}∗ × {0, 1}∗ I a language L ⊆ {0, 1}∗ induces label decoder

(x, y) ∈ FL ⇔ xy ∈ L

I complexity class P graph class complexity class GP

C ∈ GP ⇔

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 7/16 I C ∈ GP implies polylog(n) query time

Complexity of Label Decoders

I label decoder F ⊆ {0, 1}∗ × {0, 1}∗ I a language L ⊆ {0, 1}∗ induces label decoder

(x, y) ∈ FL ⇔ xy ∈ L

I complexity class P graph class complexity class GP

C ∈ GP ⇔ C has a labeling scheme (FL, c) with L ∈ P and c ∈ N

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 7/16 Complexity of Label Decoders

I label decoder F ⊆ {0, 1}∗ × {0, 1}∗ I a language L ⊆ {0, 1}∗ induces label decoder

(x, y) ∈ FL ⇔ xy ∈ L

I complexity class P graph class complexity class GP

C ∈ GP ⇔ C has a labeling scheme (FL, c) with L ∈ P and c ∈ N

I C ∈ GP implies polylog(n) query time

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 7/16 some classes with ALS classes not known to have ALS planar graphs, line segment graphs, threshold graphs, (unit) disk graphs, line graphs, k-dot product graphs | {z } CA graphs, candidates for circle graphs, implicit graph conjecture proper minor-closed, uniformly sparse, b. boxicity, b. interval number, more restrictive b. tree-width, model of computation? b. clique-width

GALL = GAC0?

Complexity of Graph Classes

GALL

GAC0

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 8/16 classes not known to have ALS planar graphs, line segment graphs, threshold graphs, (unit) disk graphs, line graphs, k-dot product graphs | {z } CA graphs, candidates for circle graphs, implicit graph conjecture proper minor-closed, uniformly sparse, b. boxicity, b. interval number, more restrictive b. tree-width, model of computation? b. clique-width

GALL = GAC0?

Complexity of Graph Classes

some classes with ALS

GALL

GAC0

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 8/16 classes not known to have ALS line segment graphs, (unit) disk graphs, k-dot product graphs | {z } candidates for implicit graph conjecture

more restrictive model of computation?

GALL = GAC0?

Complexity of Graph Classes

some classes with ALS planar graphs, threshold graphs, GALL line graphs, CA graphs, circle graphs, proper minor-closed, uniformly sparse, GP b. boxicity, b. interval number, b. tree-width, b. clique-width

GAC0

more restrictive model of computation?

GALL = GAC0?

Complexity of Graph Classes

GAC0

more restrictive model of computation?

GALL = GAC0?

Complexity of Graph Classes

GAC0

more restrictive model of computation?

GALL = GAC0?

Complexity of Graph Classes

GAC0

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 8/16 line segment graphs, (unit) disk graphs, k-dot product graphs | {z } candidates for implicit graph conjecture

more restrictive model of computation?

GALL = GAC0?

Complexity of Graph Classes

some classes with ALS classes not known to have ALS planar graphs, threshold graphs, GALL line graphs, CA graphs, circle graphs, proper minor-closed, uniformly sparse, GP b. boxicity, b. interval number, b. tree-width, b. clique-width

GAC0

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 8/16 | {z } candidates for implicit graph conjecture

more restrictive model of computation?

GALL = GAC0?

Complexity of Graph Classes

some classes with ALS classes not known to have ALS planar graphs, line segment graphs, threshold graphs, GALL (unit) disk graphs, line graphs, k-dot product graphs CA graphs, circle graphs, proper minor-closed, uniformly sparse, GP b. boxicity, b. interval number, b. tree-width, b. clique-width

GAC0

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 8/16 more restrictive model of computation?

GALL = GAC0?

Complexity of Graph Classes

some classes with ALS classes not known to have ALS planar graphs, line segment graphs, threshold graphs, GALL (unit) disk graphs, line graphs, k-dot product graphs | {z } CA graphs, candidates for circle graphs, implicit graph conjecture proper minor-closed, uniformly sparse, GP b. boxicity, b. interval number, b. tree-width, b. clique-width

GAC0

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 8/16 GALL = GAC0?

Complexity of Graph Classes

GAC0

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 8/16 Complexity of Graph Classes

GALL = GAC0? GAC0

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 8/16 ⇒ D ∈/ GEXP

I ﬁnd class D ∈ G2EXP \ GEXP I GEXP = [{C1, C2,... }]⊆ I D = {G1, G2,..., } with Gi ∈/ Ci

every hereditary graph class with factorial speed is in GP

I proof via diagonalization

0 I ⇒ GAC 6= GALL I implicit graph conjecture (Kannan et al ’92, Spinrad ’03):

Hierarchy of Complexity Classes

I hierarchy theorem (C. ’16, Rotbart & Simonsen ’16):

GP ⊆ GEXP ( G2EXP ( ··· ( GR ( GALL

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 9/16 ⇒ D ∈/ GEXP

every hereditary graph class with factorial speed is in GP

I ﬁnd class D ∈ G2EXP \ GEXP I GEXP = [{C1, C2,... }]⊆ I D = {G1, G2,..., } with Gi ∈/ Ci 0 I ⇒ GAC 6= GALL I implicit graph conjecture (Kannan et al ’92, Spinrad ’03):

Hierarchy of Complexity Classes

I hierarchy theorem (C. ’16, Rotbart & Simonsen ’16):

GP ⊆ GEXP ( G2EXP ( ··· ( GR ( GALL

I proof via diagonalization

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 9/16 ⇒ D ∈/ GEXP

every hereditary graph class with factorial speed is in GP

I GEXP = [{C1, C2,... }]⊆ I D = {G1, G2,..., } with Gi ∈/ Ci 0 I ⇒ GAC 6= GALL I implicit graph conjecture (Kannan et al ’92, Spinrad ’03):

Hierarchy of Complexity Classes

I hierarchy theorem (C. ’16, Rotbart & Simonsen ’16):

GP ⊆ GEXP ( G2EXP ( ··· ( GR ( GALL

I proof via diagonalization I ﬁnd class D ∈ G2EXP \ GEXP

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 9/16 ⇒ D ∈/ GEXP

every hereditary graph class with factorial speed is in GP

I D = {G1, G2,..., } with Gi ∈/ Ci 0 I ⇒ GAC 6= GALL I implicit graph conjecture (Kannan et al ’92, Spinrad ’03):

Hierarchy of Complexity Classes

I hierarchy theorem (C. ’16, Rotbart & Simonsen ’16):

GP ⊆ GEXP ( G2EXP ( ··· ( GR ( GALL

I proof via diagonalization I ﬁnd class D ∈ G2EXP \ GEXP I GEXP = [{C1, C2,... }]⊆

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 9/16 every hereditary graph class with factorial speed is in GP

⇒ D ∈/ GEXP 0 I ⇒ GAC 6= GALL I implicit graph conjecture (Kannan et al ’92, Spinrad ’03):

Hierarchy of Complexity Classes

I hierarchy theorem (C. ’16, Rotbart & Simonsen ’16):

GP ⊆ GEXP ( G2EXP ( ··· ( GR ( GALL

I proof via diagonalization I ﬁnd class D ∈ G2EXP \ GEXP I GEXP = [{C1, C2,... }]⊆ I D = {G1, G2,..., } with Gi ∈/ Ci

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 9/16 every hereditary graph class with factorial speed is in GP

0 I ⇒ GAC 6= GALL I implicit graph conjecture (Kannan et al ’92, Spinrad ’03):

Hierarchy of Complexity Classes

I hierarchy theorem (C. ’16, Rotbart & Simonsen ’16):

GP ⊆ GEXP ( G2EXP ( ··· ( GR ( GALL

I proof via diagonalization I ﬁnd class D ∈ G2EXP \ GEXP I GEXP = [{C1, C2,... }]⊆ I D = {G1, G2,..., } with Gi ∈/ Ci ⇒ D ∈/ GEXP

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 9/16 every hereditary graph class with factorial speed is in GP

I implicit graph conjecture (Kannan et al ’92, Spinrad ’03):

Hierarchy of Complexity Classes

I hierarchy theorem (C. ’16, Rotbart & Simonsen ’16):

GP ⊆ GEXP ( G2EXP ( ··· ( GR ( GALL

I proof via diagonalization I ﬁnd class D ∈ G2EXP \ GEXP I GEXP = [{C1, C2,... }]⊆ I D = {G1, G2,..., } with Gi ∈/ Ci ⇒ D ∈/ GEXP 0 I ⇒ GAC 6= GALL

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 9/16 every hereditary graph class with factorial speed is in GP

Hierarchy of Complexity Classes

I hierarchy theorem (C. ’16, Rotbart & Simonsen ’16):

GP ⊆ GEXP ( G2EXP ( ··· ( GR ( GALL

I proof via diagonalization I ﬁnd class D ∈ G2EXP \ GEXP I GEXP = [{C1, C2,... }]⊆ I D = {G1, G2,..., } with Gi ∈/ Ci ⇒ D ∈/ GEXP 0 I ⇒ GAC 6= GALL I implicit graph conjecture (Kannan et al ’92, Spinrad ’03):

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 9/16 Hierarchy of Complexity Classes

I hierarchy theorem (C. ’16, Rotbart & Simonsen ’16):

GP ⊆ GEXP ( G2EXP ( ··· ( GR ( GALL

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 9/16 I ϕ: FO formula using =, <, +, × I k: number of variables per vertex I c: range of variables {0, 1,..., nc} 2 v1, v2 ∈ {0,..., n }

I interval graphs in GFOqf (<)

= (1, 3) = (2, 7)

I logical labeling scheme (ϕ, k, c)

I LLS for intv. graphs: (ϕIntv, 2, 2) I classes GFO(σ) and GFOqf (σ) for σ ⊆ {<, +, ×}

v 2 7 u 1 3 4 5 6 8 9 10

`(u) = (u1, u2) ϕIntv(u , u , v , v ) ¬(u < v ∨ v < u ) `(v) = (v1, v2) 1 2 1 2 , 2 1 2 1

Logical Labeling Schemes

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 10/16 I ϕ: FO formula using =, <, +, × I k: number of variables per vertex I c: range of variables {0, 1,..., nc} 2 v1, v2 ∈ {0,..., n }

I interval graphs in GFOqf (<)

= (1, 3) = (2, 7)

I logical labeling scheme (ϕ, k, c)

I LLS for intv. graphs: (ϕIntv, 2, 2) I classes GFO(σ) and GFOqf (σ) for σ ⊆ {<, +, ×}

v 2 7 u 1 3 4 5 6 8 9 10

`(u) = (u1, u2) ϕIntv(u , u , v , v ) ¬(u < v ∨ v < u ) `(v) = (v1, v2) 1 2 1 2 , 2 1 2 1

Logical Labeling Schemes

I interval graphs in GFOqf (<)

= (1, 3) = (2, 7)

I logical labeling scheme (ϕ, k, c)

I LLS for intv. graphs: (ϕIntv, 2, 2) I classes GFO(σ) and GFOqf (σ) for σ ⊆ {<, +, ×}

`(u) = (u1, u2) ϕIntv(u , u , v , v ) ¬(u < v ∨ v < u ) `(v) = (v1, v2) 1 2 1 2 , 2 1 2 1

Logical Labeling Schemes

2 7 1 3 4 5 6 8 9 10

I interval graphs in GFOqf (<)

= (1, 3) = (2, 7)

I logical labeling scheme (ϕ, k, c)

I LLS for intv. graphs: (ϕIntv, 2, 2) I classes GFO(σ) and GFOqf (σ) for σ ⊆ {<, +, ×}

`(u) = (u1, u2) ϕIntv(u , u , v , v ) ¬(u < v ∨ v < u ) `(v) = (v1, v2) 1 2 1 2 , 2 1 2 1

Logical Labeling Schemes

v 2 7 u 1 3 4 5 6 8 9 10

I interval graphs in GFOqf (<)

I logical labeling scheme (ϕ, k, c)

I LLS for intv. graphs: (ϕIntv, 2, 2) I classes GFO(σ) and GFOqf (σ) for σ ⊆ {<, +, ×}

= (1, 3) = (2, 7) ϕIntv(u1, u2, v1, v2) , ¬(u2 < v1 ∨ v2 < u1)

Logical Labeling Schemes

v 2 7 u 1 3 4 5 6 8 9 10

`(u) = (u1, u2) `(v) = (v1, v2)

I interval graphs in GFOqf (<)

I logical labeling scheme (ϕ, k, c)

I LLS for intv. graphs: (ϕIntv, 2, 2) I classes GFO(σ) and GFOqf (σ) for σ ⊆ {<, +, ×}

= (2, 7) ϕIntv(u1, u2, v1, v2) , ¬(u2 < v1 ∨ v2 < u1)

Logical Labeling Schemes

v 2 7 u 1 3 4 5 6 8 9 10

`(u) = (u1, u2) = (1, 3) `(v) = (v1, v2)

I interval graphs in GFOqf (<)

I logical labeling scheme (ϕ, k, c)

I LLS for intv. graphs: (ϕIntv, 2, 2) I classes GFO(σ) and GFOqf (σ) for σ ⊆ {<, +, ×}

ϕIntv(u1, u2, v1, v2) , ¬(u2 < v1 ∨ v2 < u1)

Logical Labeling Schemes

v 2 7 u 1 3 4 5 6 8 9 10

`(u) = (u1, u2) = (1, 3) `(v) = (v1, v2) = (2, 7)

I interval graphs in GFOqf (<)

I logical labeling scheme (ϕ, k, c)

I LLS for intv. graphs: (ϕIntv, 2, 2) I classes GFO(σ) and GFOqf (σ) for σ ⊆ {<, +, ×}

Logical Labeling Schemes

v 2 7 u 1 3 4 5 6 8 9 10

`(u) = (u1, u2) = (1, 3) ϕIntv(u , u , v , v ) ¬(u < v ∨ v < u ) `(v) = (v1, v2) = (2, 7) 1 2 1 2 , 2 1 2 1

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 10/16 2 v1, v2 ∈ {0,..., n }

I interval graphs in GFOqf (<)

I ϕ: FO formula using =, <, +, × I k: number of variables per vertex I c: range of variables {0, 1,..., nc} I LLS for intv. graphs: (ϕIntv, 2, 2) I classes GFO(σ) and GFOqf (σ) for σ ⊆ {<, +, ×}

Logical Labeling Schemes

v 2 7 u 1 3 4 5 6 8 9 10

`(u) = (u1, u2) = (1, 3) ϕIntv(u , u , v , v ) ¬(u < v ∨ v < u ) `(v) = (v1, v2) = (2, 7) 1 2 1 2 , 2 1 2 1

I logical labeling scheme (ϕ, k, c)

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 10/16 2 v1, v2 ∈ {0,..., n }

I interval graphs in GFOqf (<)

I k: number of variables per vertex I c: range of variables {0, 1,..., nc} I LLS for intv. graphs: (ϕIntv, 2, 2) I classes GFO(σ) and GFOqf (σ) for σ ⊆ {<, +, ×}

Logical Labeling Schemes

v 2 7 u 1 3 4 5 6 8 9 10

`(u) = (u1, u2) = (1, 3) ϕIntv(u , u , v , v ) ¬(u < v ∨ v < u ) `(v) = (v1, v2) = (2, 7) 1 2 1 2 , 2 1 2 1

I logical labeling scheme (ϕ, k, c) I ϕ: FO formula using =, <, +, ×

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 10/16 2 v1, v2 ∈ {0,..., n }

I interval graphs in GFOqf (<)

I c: range of variables {0, 1,..., nc} I LLS for intv. graphs: (ϕIntv, 2, 2) I classes GFO(σ) and GFOqf (σ) for σ ⊆ {<, +, ×}

Logical Labeling Schemes

v 2 7 u 1 3 4 5 6 8 9 10

`(u) = (u1, u2) = (1, 3) ϕIntv(u , u , v , v ) ¬(u < v ∨ v < u ) `(v) = (v1, v2) = (2, 7) 1 2 1 2 , 2 1 2 1

I logical labeling scheme (ϕ, k, c) I ϕ: FO formula using =, <, +, × I k: number of variables per vertex

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 10/16 2 v1, v2 ∈ {0,..., n }

I interval graphs in GFOqf (<)

I LLS for intv. graphs: (ϕIntv, 2, 2) I classes GFO(σ) and GFOqf (σ) for σ ⊆ {<, +, ×}

Logical Labeling Schemes

v 2 7 u 1 3 4 5 6 8 9 10

`(u) = (u1, u2) = (1, 3) ϕIntv(u , u , v , v ) ¬(u < v ∨ v < u ) `(v) = (v1, v2) = (2, 7) 1 2 1 2 , 2 1 2 1

I logical labeling scheme (ϕ, k, c) I ϕ: FO formula using =, <, +, × I k: number of variables per vertex I c: range of variables {0, 1,..., nc}

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 10/16 I interval graphs in GFOqf (<)

2 v1, v2 ∈ {0,..., n } I classes GFO(σ) and GFOqf (σ) for σ ⊆ {<, +, ×}

Logical Labeling Schemes

v 2 7 u 1 3 4 5 6 8 9 10

`(u) = (u1, u2) = (1, 3) ϕIntv(u , u , v , v ) ¬(u < v ∨ v < u ) `(v) = (v1, v2) = (2, 7) 1 2 1 2 , 2 1 2 1

I logical labeling scheme (ϕ, k, c) I ϕ: FO formula using =, <, +, × I k: number of variables per vertex I c: range of variables {0, 1,..., nc} I LLS for intv. graphs: (ϕIntv, 2, 2)

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 10/16 I interval graphs in GFOqf (<)

I classes GFO(σ) and GFOqf (σ) for σ ⊆ {<, +, ×}

Logical Labeling Schemes

v 2 7 u 1 3 4 5 6 8 9 10

`(u) = (u1, u2) = (1, 3) ϕIntv(u , u , v , v ) ¬(u < v ∨ v < u ) `(v) = (v1, v2) = (2, 7) 1 2 1 2 , 2 1 2 1

I logical labeling scheme (ϕ, k, c) I ϕ: FO formula using =, <, +, × I k: number of variables per vertex I c: range of variables {0, 1,..., nc} 2 I LLS for intv. graphs: (ϕIntv, 2, 2) v1, v2 ∈ {0,..., n }

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 10/16 2 v1, v2 ∈ {0,..., n }

I interval graphs in GFOqf (<)

Logical Labeling Schemes

v 2 7 u 1 3 4 5 6 8 9 10

`(u) = (u1, u2) = (1, 3) ϕIntv(u , u , v , v ) ¬(u < v ∨ v < u ) `(v) = (v1, v2) = (2, 7) 1 2 1 2 , 2 1 2 1

I logical labeling scheme (ϕ, k, c) I ϕ: FO formula using =, <, +, × I k: number of variables per vertex I c: range of variables {0, 1,..., nc} I LLS for intv. graphs: (ϕIntv, 2, 2) I classes GFO(σ) and GFOqf (σ) for σ ⊆ {<, +, ×}

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 10/16 2 v1, v2 ∈ {0,..., n }

Logical Labeling Schemes

v 2 7 u 1 3 4 5 6 8 9 10

`(u) = (u1, u2) = (1, 3) ϕIntv(u , u , v , v ) ¬(u < v ∨ v < u ) `(v) = (v1, v2) = (2, 7) 1 2 1 2 , 2 1 2 1

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 10/16 GFO(<) = = [LNG]BF = [Interval]sg

= [Trees]sg

GPH [Factorial Speed ∩ Hereditary]⊆

GP GFO(<, +, ×)

PBS = [GFOqf(<, +, ×)]hc,⊆ GTC0

GFOqf(<, +, ×) = const. time RAM w/o division

GAC0

GFOqf(<) = GFOqf(<, +) = GFOqf(<, ×)

GFO(=) = GFOqf(=) = [Dichotomic]BF

Logical Complexity Classes

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 11/16 GFO(<) = = [LNG]BF = [Interval]sg

= [Trees]sg

GPH [Factorial Speed ∩ Hereditary]⊆

PBS = [GFOqf(<, +, ×)]hc,⊆ GTC0

GFOqf(<, +, ×) = const. time RAM w/o division

GAC0

GFOqf(<) = GFOqf(<, +) = GFOqf(<, ×)

GFO(=) = GFOqf(=) = [Dichotomic]BF

Logical Complexity Classes

GFO(<, +, ×)

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 11/16 GFO(<) = = [LNG]BF = [Interval]sg

= [Trees]sg

GPH [Factorial Speed ∩ Hereditary]⊆

PBS = [GFOqf(<, +, ×)]hc,⊆ GTC0

= const. time RAM w/o division

GAC0

GFOqf(<) = GFOqf(<, +) = GFOqf(<, ×)

GFO(=) = GFOqf(=) = [Dichotomic]BF

Logical Complexity Classes

GFO(<, +, ×)

GFOqf(<, +, ×)

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 11/16 GFO(<) = = [LNG]BF = [Interval]sg

= [Trees]sg

GPH [Factorial Speed ∩ Hereditary]⊆

PBS = [GFOqf(<, +, ×)]hc,⊆ GTC0

= const. time RAM w/o division

GAC0

GFOqf(<) = = GFOqf(<, ×)

GFO(=) = GFOqf(=) = [Dichotomic]BF

Logical Complexity Classes

GFO(<, +, ×)

GFOqf(<, +, ×)

GFOqf(<, +)

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 11/16 GFO(<) =

= [Trees]sg

GPH [Factorial Speed ∩ Hereditary]⊆

PBS = [GFOqf(<, +, ×)]hc,⊆ GTC0

= const. time RAM w/o division

GAC0

GFOqf(<) = = [LNG]BF = [Interval]sg

GFO(=) = GFOqf(=) = [Dichotomic]BF

Logical Complexity Classes

GFO(<, +, ×)

GFOqf(<, +, ×)

GFOqf(<, +) = GFOqf(<, ×)

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 11/16 = [Trees]sg

GPH [Factorial Speed ∩ Hereditary]⊆

PBS = [GFOqf(<, +, ×)]hc,⊆ GTC0

= const. time RAM w/o division

GAC0

GFO(<) = = [LNG]BF = [Interval]sg

GFO(=) = GFOqf(=) = [Dichotomic]BF

Logical Complexity Classes

GFO(<, +, ×)

GFOqf(<, +, ×)

GFOqf(<) = GFOqf(<, +) = GFOqf(<, ×)

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 11/16 = [Trees]sg

GPH [Factorial Speed ∩ Hereditary]⊆

PBS = [GFOqf(<, +, ×)]hc,⊆ GTC0

= const. time RAM w/o division

GAC0

= [LNG]BF = [Interval]sg

GFO(=) = GFOqf(=) = [Dichotomic]BF

Logical Complexity Classes

GFO(<, +, ×)

GFOqf(<, +, ×)

GFO(<) = GFOqf(<) = GFOqf(<, +) = GFOqf(<, ×)

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 11/16 = [Trees]sg

GPH [Factorial Speed ∩ Hereditary]⊆

PBS = [GFOqf(<, +, ×)]hc,⊆ GTC0

= const. time RAM w/o division

GAC0

= [LNG]BF = [Interval]sg

GFO(=) = = [Dichotomic]BF

Logical Complexity Classes

GFO(<, +, ×)

GFOqf(<, +, ×)

GFO(<) = GFOqf(<) = GFOqf(<, +) = GFOqf(<, ×)

GFOqf(=)

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 11/16 = [Trees]sg

GPH [Factorial Speed ∩ Hereditary]⊆

PBS = [GFOqf(<, +, ×)]hc,⊆ GTC0

= const. time RAM w/o division

GAC0

= [LNG]BF = [Interval]sg

= [Dichotomic]BF

Logical Complexity Classes

GFO(<, +, ×)

GFOqf(<, +, ×)

GFO(<) = GFOqf(<) = GFOqf(<, +) = GFOqf(<, ×)

GFO(=) = GFOqf(=)

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 11/16 = [Trees]sg

[Factorial Speed ∩ Hereditary]⊆

PBS = [GFOqf(<, +, ×)]hc,⊆

= const. time RAM w/o division

= [LNG]BF = [Interval]sg

= [Dichotomic]BF

Logical Complexity Classes

GPH

GP GFO(<, +, ×)

GTC0

GFOqf(<, +, ×)

GAC0

GFO(<) = GFOqf(<) = GFOqf(<, +) = GFOqf(<, ×)

GFO(=) = GFOqf(=)

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 11/16 = [Trees]sg

[Factorial Speed ∩ Hereditary]⊆

PBS = [GFOqf(<, +, ×)]hc,⊆

= const. time RAM w/o division

= [LNG]BF = [Interval]sg

= [Dichotomic]BF

Logical Complexity Classes

GPH

GP GFO(<, +, ×)

GTC0

GFOqf(<, +, ×)

GAC0

GFO(<) = GFOqf(<) = GFOqf(<, +) = GFOqf(<, ×)

GFO(=) = GFOqf(=)

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 11/16 = [Trees]sg

[Factorial Speed ∩ Hereditary]⊆

PBS = [GFOqf(<, +, ×)]hc,⊆

= const. time RAM w/o division

= [LNG]BF = [Interval]sg

= [Dichotomic]BF

Logical Complexity Classes

GPH

GP GFO(<, +, ×)

GTC0

GFOqf(<, +, ×)

GAC0

GFO(<) = GFOqf(<) = GFOqf(<, +) = GFOqf(<, ×)

GFO(=) = GFOqf(=)

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 11/16 = [Trees]sg

[Factorial Speed ∩ Hereditary]⊆

PBS = [GFOqf(<, +, ×)]hc,⊆

= const. time RAM w/o division

= [LNG]BF = [Interval]sg

= [Dichotomic]BF

Logical Complexity Classes

GPH

GP GFO(<, +, ×)

GTC0

GFOqf(<, +, ×)

GAC0

GFO(<) = GFOqf(<) = GFOqf(<, +) = GFOqf(<, ×)

GFO(=) = GFOqf(=)

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 11/16 [Factorial Speed ∩ Hereditary]⊆

PBS = [GFOqf(<, +, ×)]hc,⊆

= const. time RAM w/o division

= [LNG]BF = [Interval]sg

= [Trees]sg

Logical Complexity Classes

GPH

GP GFO(<, +, ×)

GTC0

GFOqf(<, +, ×)

GAC0

GFO(<) = GFOqf(<) = GFOqf(<, +) = GFOqf(<, ×)

GFO(=) = GFOqf(=) = [Dichotomic]BF

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 11/16 [Factorial Speed ∩ Hereditary]⊆

PBS = [GFOqf(<, +, ×)]hc,⊆

= const. time RAM w/o division

= [LNG]BF = [Interval]sg

Logical Complexity Classes

GPH

GP GFO(<, +, ×)

GTC0

GFOqf(<, +, ×)

GAC0

GFO(<) = GFOqf(<) = GFOqf(<, +) = GFOqf(<, ×)

GFO(=) = GFOqf(=) = [Dichotomic]BF = [Trees]sg

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 11/16 [Factorial Speed ∩ Hereditary]⊆

PBS = [GFOqf(<, +, ×)]hc,⊆

= const. time RAM w/o division

= [Interval]sg

Logical Complexity Classes

GPH

GP GFO(<, +, ×)

GTC0

GFOqf(<, +, ×)

GAC0

GFO(<) = GFOqf(<) = GFOqf(<, +) = GFOqf(<, ×) = [LNG]BF

GFO(=) = GFOqf(=) = [Dichotomic]BF = [Trees]sg

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 11/16 [Factorial Speed ∩ Hereditary]⊆

PBS = [GFOqf(<, +, ×)]hc,⊆

= const. time RAM w/o division

Logical Complexity Classes

GPH

GP GFO(<, +, ×)

GTC0

GFOqf(<, +, ×)

GAC0

GFO(<) = GFOqf(<) = GFOqf(<, +) = GFOqf(<, ×) = [LNG]BF = [Interval]sg

GFO(=) = GFOqf(=) = [Dichotomic]BF = [Trees]sg

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 11/16 [Factorial Speed ∩ Hereditary]⊆

PBS = [GFOqf(<, +, ×)]hc,⊆

Logical Complexity Classes

GPH

GP GFO(<, +, ×)

GTC0

GFOqf(<, +, ×) = const. time RAM w/o division

GAC0

GFO(<) = GFOqf(<) = GFOqf(<, +) = GFOqf(<, ×) = [LNG]BF = [Interval]sg

GFO(=) = GFOqf(=) = [Dichotomic]BF = [Trees]sg

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 11/16 [Factorial Speed ∩ Hereditary]⊆

= [GFOqf(<, +, ×)]hc,⊆

Logical Complexity Classes

GPH

GP GFO(<, +, ×) PBS GTC0

GFOqf(<, +, ×) = const. time RAM w/o division

GAC0

GFO(<) = GFOqf(<) = GFOqf(<, +) = GFOqf(<, ×) = [LNG]BF = [Interval]sg

GFO(=) = GFOqf(=) = [Dichotomic]BF = [Trees]sg

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 11/16 = [GFOqf(<, +, ×)]hc,⊆

Logical Complexity Classes

GPH [Factorial Speed ∩ Hereditary]⊆

GP GFO(<, +, ×) PBS GTC0

GFOqf(<, +, ×) = const. time RAM w/o division

GAC0

GFO(<) = GFOqf(<) = GFOqf(<, +) = GFOqf(<, ×) = [LNG]BF = [Interval]sg

GFO(=) = GFOqf(=) = [Dichotomic]BF = [Trees]sg

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 11/16 = [GFOqf(<, +, ×)]hc,⊆

Logical Complexity Classes

GPH [Factorial Speed ∩ Hereditary]⊆

GP GFO(<, +, ×) PBS GTC0

GFOqf(<, +, ×) = const. time RAM w/o division

GAC0

GFO(<) = GFOqf(<) = GFOqf(<, +) = GFOqf(<, ×) = [LNG]BF = [Interval]sg

GFO(=) = GFOqf(=) = [Dichotomic]BF = [Trees]sg

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 11/16 Logical Complexity Classes

GPH [Factorial Speed ∩ Hereditary]⊆

GP GFO(<, +, ×)

PBS = [GFOqf(<, +, ×)]hc,⊆ GTC0

GFOqf(<, +, ×) = const. time RAM w/o division

GAC0

GFO(<) = GFOqf(<) = GFOqf(<, +) = GFOqf(<, ×) = [LNG]BF = [Interval]sg

GFO(=) = GFOqf(=) = [Dichotomic]BF = [Trees]sg

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 11/16 some classes with ALS planar graphs, threshold graphs, clique-width ⊆ PBS? line graphs, CA graphs, circle graphs, proper minor-closed, uniformly sparse, b. boxicity, b. interval number, b. tree-width, GFO(<) contains everything* but clique-width

Reﬁned Complexity of Graph Classes

GALL

GAC0

GFO(<)

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 12/16 planar graphs, threshold graphs, clique-width ⊆ PBS? line graphs, CA graphs, circle graphs, proper minor-closed, uniformly sparse, b. boxicity, b. interval number, b. tree-width, GFO(<) contains everything* but clique-width

Reﬁned Complexity of Graph Classes

some classes with ALS

GALL

GAC0

GFO(<)

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 12/16 clique-width ⊆ PBS?

GFO(<) contains everything* but clique-width

Reﬁned Complexity of Graph Classes

some classes with ALS planar graphs, threshold graphs, GALL line graphs, CA graphs, circle graphs, proper minor-closed, 0 uniformly sparse, GAC b. boxicity, b. interval number, b. tree-width, b. clique-width

GFO(<)

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 12/16 clique-width ⊆ PBS?

GFO(<) contains everything* but clique-width

Reﬁned Complexity of Graph Classes

GFO(<)

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 12/16 clique-width ⊆ PBS?

GFO(<) contains everything* but clique-width

Reﬁned Complexity of Graph Classes

GFO(<)

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 12/16 clique-width ⊆ PBS?

GFO(<) contains everything* but clique-width

Reﬁned Complexity of Graph Classes

GFO(<)

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 12/16 clique-width ⊆ PBS?

GFO(<) contains everything* but clique-width

Reﬁned Complexity of Graph Classes

GFO(<)

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 12/16 clique-width ⊆ PBS?

Reﬁned Complexity of Graph Classes

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 12/16 Reﬁned Complexity of Graph Classes

some classes with ALS planar graphs, threshold graphs, GALL clique-width ⊆ PBS? line graphs, CA graphs, circle graphs, proper minor-closed, 0 uniformly sparse, GAC b. boxicity, b. interval number, b. tree-width, b. clique-width GFO(<) contains everything* but GFO(<) clique-width

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 12/16 if C ≤ D and D ∈ GP then C ∈ GP I express graphs in C with graphs in D

I box graphs ≤BF interval graphs

2 2 2 1 3 = 1 3 ∧ 1 3 4 4 4

Reduction for Graph Classes

I closure:

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 13/16 then C ∈ GP I express graphs in C with graphs in D

I box graphs ≤BF interval graphs

2 2 2 1 3 = 1 3 ∧ 1 3 4 4 4

Reduction for Graph Classes

I closure: if C ≤ D and D ∈ GP

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 13/16 I express graphs in C with graphs in D

I box graphs ≤BF interval graphs

2 2 2 1 3 = 1 3 ∧ 1 3 4 4 4

Reduction for Graph Classes

I closure: if C ≤ D and D ∈ GP then C ∈ GP

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 13/16 I box graphs ≤BF interval graphs

2 2 2 1 3 = 1 3 ∧ 1 3 4 4 4

Reduction for Graph Classes

I closure: if C ≤ D and D ∈ GP then C ∈ GP I express graphs in C with graphs in D

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 13/16 I box graphs ≤BF interval graphs

2 2 = 1 3 ∧ 1 3 4 4

Reduction for Graph Classes

I closure: if C ≤ D and D ∈ GP then C ∈ GP I express graphs in C with graphs in D

1 3

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 13/16 I box graphs ≤BF interval graphs

Reduction for Graph Classes

I closure: if C ≤ D and D ∈ GP then C ∈ GP I express graphs in C with graphs in D

2 2 2 1 3 = 1 3 ∧ 1 3 4 4 4

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 13/16 Reduction for Graph Classes

I closure: if C ≤ D and D ∈ GP then C ∈ GP I express graphs in C with graphs in D

I box graphs ≤BF interval graphs

2 2 2 1 3 = 1 3 ∧ 1 3 4 4 4

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 13/16 planar ∧ planar = planar

⇒ box graphs ≤BF interval

¬C := co-C C ∨ D := {G | ∃H ∈ C, J ∈ D : E(G) = E(H) ∪ E(J)} I C ∧ D = ¬(¬C ∨ ¬D)

I C ≤BF D if ∃ boolean formula F: C ⊆ F(D,..., D) I box graphs ⊆ interval ∧ interval

Algebraic Reduction ≤BF

I boolean formulas for graph classes:

2 2 2 1 3 = 1 3 ∧ 1 3 4 4 4

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 14/16 planar ∧ planar = planar

⇒ box graphs ≤BF interval

C ∨ D := {G | ∃H ∈ C, J ∈ D : E(G) = E(H) ∪ E(J)} I C ∧ D = ¬(¬C ∨ ¬D)

I C ≤BF D if ∃ boolean formula F: C ⊆ F(D,..., D) I box graphs ⊆ interval ∧ interval

Algebraic Reduction ≤BF

I boolean formulas for graph classes: ¬C := co-C

2 2 2 1 3 = 1 3 ∧ 1 3 4 4 4

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 14/16 planar ∧ planar = planar

⇒ box graphs ≤BF interval

I C ∧ D = ¬(¬C ∨ ¬D)

I C ≤BF D if ∃ boolean formula F: C ⊆ F(D,..., D) I box graphs ⊆ interval ∧ interval

Algebraic Reduction ≤BF

I boolean formulas for graph classes: ¬C := co-C C ∨ D := {G | ∃H ∈ C, J ∈ D : E(G) = E(H) ∪ E(J)}

2 2 2 1 3 = 1 3 ∧ 1 3 4 4 4

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 14/16 ⇒ box graphs ≤BF interval

planar ∧ planar = planar

I C ≤BF D if ∃ boolean formula F: C ⊆ F(D,..., D) I box graphs ⊆ interval ∧ interval

Algebraic Reduction ≤BF

I boolean formulas for graph classes: ¬C := co-C C ∨ D := {G | ∃H ∈ C, J ∈ D : E(G) = E(H) ∪ E(J)} I C ∧ D = ¬(¬C ∨ ¬D)

2 2 2 1 3 = 1 3 ∧ 1 3 4 4 4

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 14/16 ⇒ box graphs ≤BF interval

planar

I C ≤BF D if ∃ boolean formula F: C ⊆ F(D,..., D) I box graphs ⊆ interval ∧ interval

Algebraic Reduction ≤BF

I boolean formulas for graph classes: ¬C := co-C C ∨ D := {G | ∃H ∈ C, J ∈ D : E(G) = E(H) ∪ E(J)} I C ∧ D = ¬(¬C ∨ ¬D) planar ∧ planar =

2 2 2 1 3 = 1 3 ∧ 1 3 4 4 4

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 14/16 ⇒ box graphs ≤BF interval

I C ≤BF D if ∃ boolean formula F: C ⊆ F(D,..., D) I box graphs ⊆ interval ∧ interval

Algebraic Reduction ≤BF

I boolean formulas for graph classes: ¬C := co-C C ∨ D := {G | ∃H ∈ C, J ∈ D : E(G) = E(H) ∪ E(J)} I C ∧ D = ¬(¬C ∨ ¬D) planar ∧ planar = planar

2 2 2 1 3 = 1 3 ∧ 1 3 4 4 4

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 14/16 ⇒ box graphs ≤BF interval

I box graphs ⊆ interval ∧ interval

Algebraic Reduction ≤BF

I boolean formulas for graph classes: ¬C := co-C C ∨ D := {G | ∃H ∈ C, J ∈ D : E(G) = E(H) ∪ E(J)} I C ∧ D = ¬(¬C ∨ ¬D) planar ∧ planar = planar

I C ≤BF D if ∃ boolean formula F: C ⊆ F(D,..., D)

2 2 2 1 3 = 1 3 ∧ 1 3 4 4 4

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 14/16 ⇒ box graphs ≤BF interval

Algebraic Reduction ≤BF

I boolean formulas for graph classes: ¬C := co-C C ∨ D := {G | ∃H ∈ C, J ∈ D : E(G) = E(H) ∪ E(J)} I C ∧ D = ¬(¬C ∨ ¬D) planar ∧ planar = planar

I C ≤BF D if ∃ boolean formula F: C ⊆ F(D,..., D) I box graphs ⊆ interval ∧ interval

2 2 2 1 3 = 1 3 ∧ 1 3 4 4 4

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 14/16 Algebraic Reduction ≤BF

I boolean formulas for graph classes: ¬C := co-C C ∨ D := {G | ∃H ∈ C, J ∈ D : E(G) = E(H) ∪ E(J)} I C ∧ D = ¬(¬C ∨ ¬D) planar ∧ planar = planar

I C ≤BF D if ∃ boolean formula F: C ⊆ F(D,..., D)

I box graphs ⊆ interval ∧ interval ⇒ box graphs ≤BF interval

2 2 2 1 3 = 1 3 ∧ 1 3 4 4 4

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 14/16 C ≤BF D ∧ D ∈ X ⇒ C ∈ X

C ≤BF forest ⊆ interval ∧ intverval

I every complexity class X shown here is closed under ≤BF

I GFO(=), GFO(<) have ≤BF-complete graph classes 0 I GAC has no hereditary ≤BF-complete graph class I Uniformly Sparse (US) = [Sparse ∩ Hereditary]⊆

I ∀C ∈ US: C ≤BF interval

More about ≤BF

I C ⊆ D ⇒ C ≤BF D

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 15/16 C ≤BF forest ⊆ interval ∧ intverval

C ≤BF D ∧ D ∈ X ⇒ C ∈ X

I GFO(=), GFO(<) have ≤BF-complete graph classes 0 I GAC has no hereditary ≤BF-complete graph class I Uniformly Sparse (US) = [Sparse ∩ Hereditary]⊆

I ∀C ∈ US: C ≤BF interval

More about ≤BF

I C ⊆ D ⇒ C ≤BF D

I every complexity class X shown here is closed under ≤BF

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 15/16 C ≤BF forest ⊆ interval ∧ intverval

I GFO(=), GFO(<) have ≤BF-complete graph classes 0 I GAC has no hereditary ≤BF-complete graph class I Uniformly Sparse (US) = [Sparse ∩ Hereditary]⊆

I ∀C ∈ US: C ≤BF interval

More about ≤BF

I C ⊆ D ⇒ C ≤BF D

I every complexity class X shown here is closed under ≤BF

C ≤BF D ∧ D ∈ X ⇒ C ∈ X

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 15/16 C ≤BF forest ⊆ interval ∧ intverval

0 I GAC has no hereditary ≤BF-complete graph class I Uniformly Sparse (US) = [Sparse ∩ Hereditary]⊆

I ∀C ∈ US: C ≤BF interval

More about ≤BF

I C ⊆ D ⇒ C ≤BF D

I every complexity class X shown here is closed under ≤BF

C ≤BF D ∧ D ∈ X ⇒ C ∈ X

I GFO(=), GFO(<) have ≤BF-complete graph classes

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 15/16 C ≤BF forest ⊆ interval ∧ intverval

I Uniformly Sparse (US) = [Sparse ∩ Hereditary]⊆

I ∀C ∈ US: C ≤BF interval

More about ≤BF

I C ⊆ D ⇒ C ≤BF D

I every complexity class X shown here is closed under ≤BF

C ≤BF D ∧ D ∈ X ⇒ C ∈ X

I GFO(=), GFO(<) have ≤BF-complete graph classes 0 I GAC has no hereditary ≤BF-complete graph class

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 15/16 C ≤BF forest ⊆ interval ∧ intverval

I ∀C ∈ US: C ≤BF interval

More about ≤BF

I C ⊆ D ⇒ C ≤BF D

I every complexity class X shown here is closed under ≤BF

C ≤BF D ∧ D ∈ X ⇒ C ∈ X

I GFO(=), GFO(<) have ≤BF-complete graph classes 0 I GAC has no hereditary ≤BF-complete graph class I Uniformly Sparse (US) = [Sparse ∩ Hereditary]⊆

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 15/16 C ≤BF forest ⊆ interval ∧ intverval

More about ≤BF

I C ⊆ D ⇒ C ≤BF D

I every complexity class X shown here is closed under ≤BF

C ≤BF D ∧ D ∈ X ⇒ C ∈ X

I GFO(=), GFO(<) have ≤BF-complete graph classes 0 I GAC has no hereditary ≤BF-complete graph class I Uniformly Sparse (US) = [Sparse ∩ Hereditary]⊆

I ∀C ∈ US: C ≤BF interval

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 15/16 ⊆ interval ∧ intverval

More about ≤BF

I C ⊆ D ⇒ C ≤BF D

I every complexity class X shown here is closed under ≤BF

C ≤BF D ∧ D ∈ X ⇒ C ∈ X

I GFO(=), GFO(<) have ≤BF-complete graph classes 0 I GAC has no hereditary ≤BF-complete graph class I Uniformly Sparse (US) = [Sparse ∩ Hereditary]⊆

I ∀C ∈ US: C ≤BF interval

C ≤BF forest

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 15/16 More about ≤BF

I C ⊆ D ⇒ C ≤BF D

I every complexity class X shown here is closed under ≤BF

C ≤BF D ∧ D ∈ X ⇒ C ∈ X

I GFO(=), GFO(<) have ≤BF-complete graph classes 0 I GAC has no hereditary ≤BF-complete graph class I Uniformly Sparse (US) = [Sparse ∩ Hereditary]⊆

I ∀C ∈ US: C ≤BF interval

C ≤BF forest ⊆ interval ∧ intverval

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 15/16 I GFO(=) = GFO(<) = GFOqf(<, +, ×) = PBS = [Factorial ∩ Hereditary]⊆ ? I complete graph class for GFO(=) restricted to undirected graph classes?

I disk graphs ≤BF line segment graphs? I generalized ALS for planar graphs → or-pointer number (OPN) OPN = GFO(=) restricted to undir. classes?

Research Questions

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 16/16 I complete graph class for GFO(=) restricted to undirected graph classes?

I disk graphs ≤BF line segment graphs? I generalized ALS for planar graphs → or-pointer number (OPN) OPN = GFO(=) restricted to undir. classes?

Research Questions

I GFO(=) = GFO(<) = GFOqf(<, +, ×) = PBS = [Factorial ∩ Hereditary]⊆ ?

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 16/16 I disk graphs ≤BF line segment graphs? I generalized ALS for planar graphs → or-pointer number (OPN) OPN = GFO(=) restricted to undir. classes?

Research Questions

I GFO(=) = GFO(<) = GFOqf(<, +, ×) = PBS = [Factorial ∩ Hereditary]⊆ ? I complete graph class for GFO(=) restricted to undirected graph classes?

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 16/16 I generalized ALS for planar graphs → or-pointer number (OPN) OPN = GFO(=) restricted to undir. classes?

Research Questions

I GFO(=) = GFO(<) = GFOqf(<, +, ×) = PBS = [Factorial ∩ Hereditary]⊆ ? I complete graph class for GFO(=) restricted to undirected graph classes?

I disk graphs ≤BF line segment graphs?

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 16/16 Research Questions

I GFO(=) = GFO(<) = GFOqf(<, +, ×) = PBS = [Factorial ∩ Hereditary]⊆ ? I complete graph class for GFO(=) restricted to undirected graph classes?

I disk graphs ≤BF line segment graphs? I generalized ALS for planar graphs → or-pointer number (OPN) OPN = GFO(=) restricted to undir. classes?

Maurice Chandoo Descriptive Complexity of Graph Classes via Labeling Schemes 16/16