G P P (A G ) (A G ) P (A G )

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Garey Sutton
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3 G P P (A G ) A G G (A G ) P (A G ) P (A G ) (A G ) (A G ) A G P (A G ) (A G ) (A G ) A G G A G i, j A G i j C = {0, 1,..., k} i j c > 0 c v k k + 1 k = 4 k = R(4, 3, 3) 30 n {1,..., n} true false

4 G 1 G 2 G A G1 A G2 A G3 G = (V, E) V = {1,..., n} E V V (x, y) E (y, x) E A G G n n A G [x, y] (x, y) E i th A A[i] A[i, j] j th A[i] u V degree(u) = {(u, v) (u, v) E} G δ(g) (G) δ {1,..., 7} 1 G = (V, E) n A G G π {1,..., n} π(g) G π π(g) = (V, E ) E = {(π(x), π(y)) (x, y) E} π(a G ) π(g) G G π A G = π(a G ) G 1 G 2 π 1 = (2, 8, 5, 9, 4, 7, 3) G 1 G 3 π 2 = (2, 9, 4, 8, 6, 7, 3)

5 π = (1, 2, 3)(4, 5, 6)(7, 10, 11) A A[i]A[j] i j s s n m A B A[1]A[2] A[n] B[1]B[2]... B[m] G G A G A G G G can(g) = min {π(g) π } G G = can(g) G 3 G 1 G 2 G 3 G 2 {1, 2, 3, 4} {5, 6, 7, 8} G 3

6 1 A n n l (A) = n 1 i=1 A[i] A[i + 1] l (A G1 ) = false l (A G2 ) = false l (A G3 ) = true A l (A ) A A A[i] A[i+1] G A G [1] A G [2] l (can(a)) G l (can(a G )) A A l (A) i A[i] A[i + 1] j 1 j < j A[i, j ] = A[i + 1, j ] A[i, j] > A[i + 1, j] B i, i + 1 i, i + 1 B A A A[i, j] > 0 i j j A[j, i + 1] A[j, i] > B[j, i] B A j > i A[i, 1] A[i, j 1] = B[i, 1] B[i, j 1] A[i ] = B[i ] 1 i B[i, j] B A l (A) A 1 l (A 1 ) = true A 1 [2] A 1 [3] 2

7 j i i i+1 3 A 2 l (A 2 ) = true A 2 A 1 A 2 s I {1,..., s } (s I) s I I I max(i) s 1 I s 2 (s 1 I) (s 2 I)

8 A 1 A 2 A n n l (A) = i<j A[i] {i,j} A[j] A n n l (A) l (A) l (A) i A[i] A[i + 1] A[i] {i,i+1} A[i+1] A[i, 1] A[i, i 1] = A[i+1, 1] A[i+ 1, i 1] A[i, i + 2] A[i, n] A[i + 1, i + 2] A[i + 1, n] A[i, i] = 0 A[i, i+1] = A[i+1, i] A[i, i] < A[i+1, i] A[i, i+1] = A[i+1, i] = 0 A[i] A[i + 1] O(n 2 ) O(n) s 1 {i,j} s 2 s 2 {j,k} s 3 s 1 {i,k} s 3 A A[1] {1,2} A[2] A[2] {2,4} A[4] A[1] {1,4} A[4] A[i] {i,i+1} A[i+1] A[i+1] {i+1,i+2} A[i+2] A[i] {i,i+2} A[i + 2] i i + 1 i + 2 A[i] = S 1 0 x y T 1 A[i + 1] = S 2 x 0 z T 2 A[i + 2] = S 3 y z 0 T 3 I S 1 yt 1 S 2 zt 2 S 2 xt 2 S 3 yt 3 S 1 xt 1 S 3 zt 3

9 A[1] {1,2} A[2] A[2] {2,4} A[4] A[1] {1,4} A[4] S 1 S 2 S 2 S 3 S 1 S 3 S 1 = S 2 S 2 S 3 S 2 = S 3 xt 1 zt 3 y z x y x z y = z x y x z x = y = z T 1 T 2 T 2 T 3 i j k i j i j j k j k i k i j k A[i] = S 1 0 T 1 x U 1 y V 1 A[j] = S 2 x T 2 0 U 2 z V 2 A[k] = S 3 y T 3 z U 3 0 V 3 S 1 = S 2 = S 3 T 3 T 1 T 2 x = 0 y = 1 S 1 T 1 U 1 yv 1 S 2 T 2 U 2 zv 2 S 2 xt 2 U 2 V 2 S 3 yt 3 U 3 V 3 A[i] {i,j} A[j] A[j] {j,k} A[k] S 1 T 1 xu 1 V 1 S 3 T 3 zu 3 V 3 A[i] {i,k} A[k] O(n) l (A) = A[i] {i,j} A[j] i < j j i 2 l A l (A) A G A l (A) i j i < j A[i] {i,j} A[j] π

10 1 k A[j, k] 3 k j i j A[i, j] A[j, i] k B[k, i] B A i < k < j A[i, 1] A[i, k 1] = A[j, 1] A[j, k 1] i A[1, i] A[k 1, i] = B[j, 1] B[j, k 1] i i B[i, k] k > j A[i, 1] A[i, k 1] = A[j, 1] A[j, k 1] i j A[1, i] A[k 1, i] = B[1, j] B[k 1, j] i j i B[i, k]

11 i < k < j i j j < k i j l (A) A G l (A) π i j G A A π(g) B = A π(g) i < j B A l (A) A[i] = S 1 0 S 2 x S 3 A[j] = T 1 x T 2 0 T 3 i j B[i] = T 1 0 T 2 x T 3 B[j] = S 1 x S 2 0 S 3 B

12 1 A A B A B A B S 1,, S 3 T 1,, T 3 B A k l B[k] A[k] B k l i j l = i l = j A[k, i] B[k, j] A[k, j] B[k, j] l = i k i j k = i i j A B j i A B k j k j B[k] A[k] B[i, l] < A[i, l] B[i, l] = A[j, l] A[j, l] < A[i, l] l A[i] A[j] A[i] {i,j} A[j] G P = {P 1,..., P p } G 1 i < j p v i P i v j P j v i < v j

13 P = {P 1,..., P p } G π G P 1 i p, v i P i, π(v i ) P i G 2 P = {{1, 2, 3, 4}, {5, 6, 7, 8}, {9}} π = (2, 3, 4) P P 1 P 1 G P can(g, P ) = min {π(g) π P } G P G = can(g, P ) A n n P = {P 1, P 2,..., P p } p l (A, P ) = A[i] {i,j} A[j] k=1 {i, j} P k, i < j j i 2 G P l (A G, P ) A G A l (A, P ) P k {i, j} P k i < j A[i] {i,j} A[j] B = A π(g) π i j π P B A G G 2 P = {{1, 2, 3, 4}, {5, 6, 7, 8}, {9}} G P G l (A G, P ) l (A G )

14 1 1 i<j v. (A[i, j] = A[j, i] A[i, i] = 0) i,j,k. A[i, j] + A[j, k] + A[k, i] < 3 i,j,k,l. A[i, j] + A[j, k] + A[k, l] + A[l, i] < 4 A[i, j] = e 1 i<j v 1 i v.δ A[i], δ = min i ( A[i]) = max( A[i]) i v e F k (v) v k+1 f k (v) F k (v) F k (v) f k (v) F k (v) F k (v) f k (v) F k (v) f 4 (v) v v 30 f 4 (v) 40 v 49 F 4 (v) v 21 f 4 (v) 31 v v 42 f 4 (v) F 4 (v) v e A v v e i, j, k i < k x i,j,k A[i, j] A[j, k] i k j x i,k {x i,j,k j i, j k} i k i,k. A[i, k] + x i,k < 2 i,k. j x i,j,k < 2

15 F 4 (v) e δ v 1 v 1 + δ 1 + δ 2 δ e f 4 (v 1) 2e/v v e (δ, ) l l v = 31 e = 80 (δ, ) {(4, 6), (4, 7), (5, 6)} v = 31 e = 81 (5, 6) v < 20 l 4.7 F 4 (15) 1.8 v 60 F 4 (20) f 4 (v) f 4 (v) l l l v f 4 (v) 4 (v) 4 (v) + 1

16 1 l l l + 4 f 4 (v) F(v) (δ, ) (, δ 1) (m, n) S m,n m n 1 S,δ S,δ 1 S,δ 1 δ 1 S,δ 1 S 6, S 6,4

17 S 6,4 F 4 (31) f 4 (31) f 4 (32) f 4 (31) 80 f 4 (31) = 80 f 4 (31) = 80 f 4 (32) 85 f 4 (32) 85 f 4 (32) = 85 l l F 4 (v) v v e = f 4 (v) v 15 v f 4 (v) F 4 (v) sols time

18 1 16 v 19 (, δ 1) 17 v 19 l l l l P k {i, j} P k i < j A[i] A[j] A[i] {i,j} A[j] (i, j) v = 17 (δ, ) = (4, 5) v 20 S m,[n1,...,n m] m n 1,..., n m v 21 (δ, ) F 4 (22) F 4 (23) F 4 (24) F 4 (25) F 4 (32) l l l F 4 (32) F 4 (v) 26 v 31 F 4 (v 1) F 4 (v) G v 20 v 27 v = 32 (δ, ) G G (δ, ) S m,[n1,...,n m] S m,[n 1,...,n m ] S m,[n1,...,n m] S m,[n 1,...,n m ]

19 l l F 4 (v) f 5 (v) f 5 (v) + l l

20 1 f 4(v) (δ, ) 41 (3, 5) S 5,[3,3,3,3,2] 44 (3, 5) S 5,[3,3,3,3,3] (4, 5) S 5,[3,3,3,3,3] 47 (3, 5) S 5,[4,3,3,3,3] (4, 5) S 5,[4,3,3,3,3] 50 (3, 5) S 5,[4,4,3,3,3] (4, 5) S 5,[4,4,3,3,3] S 5,[4,3,3,3,3] 54 (4, 5) S 5,[4,4,4,3,3] 57 (3, 5) S 5,[4,4,4,4,3] S 5,[4,4,4,3,3] (4, 5) S 5,[4,4,4,4,3] S 5,[4,4,4,3,3] (4, 6) S 6,[3,3,3,3,3,3] (4, 5) S 5,[4,4,4,4,4] (4, 5) S 5,[4,4,4,4,4] (3, 6) S 6,[4,4,4,4,3,2] (4, 5) S 5,[4,4,4,4,4] (4, 6) S 6,[4,4,4,3,3,3] (4, 6) S 6,[4,4,4,4,3,3] (4, 6) S 6,[4,4,4,4,4,3] S 6,[5,4,4,4,3,3] (4, 6) S 6,[4,4,4,4,4,4] S 6,[5,4,4,4,4,3] (5, 6) S 6,[4,4,4,4,4,4] (5, 6) S 6,[5,4,4,4,4,4] 20 v 25 v = 32 5 l + l l f 5 (v) l O(n 2 ) O(n) l k (G, κ) G = (V, E) κ: E {1,..., k} (G, κ)

21 1 i<j n. ( 1 A[i, j] 3 A[i, j] = A[j, i] ) 1 i n. ( A[i, i] = 0 ) 1 i<j<k n. ( ) A i,j = A j,k = A k,i (3, 3, 3; n) V = {1,..., n} n n A A[i, j] = { κ(i, j) (i, j) E 0 φ(a) A n n φ A φ A P l l 1 1 (3, 3, 3; n) K n (r 1,..., r k ; n) k K n K ri i 1 i k R(r 1,..., r k ) n > 0 (r 1,..., r k ; n) R(3, 3, 3) = 17 (3, 3, 3; n) K n n 14 n 16 (3, 3, 3; 13) (3, 3, 3; n) n 3 n K 3 (3, 3, 3; n) \

22 1 n \ (3, 3, 3; n) R(3, 3, 3; 16) (G, κ 1 ) (H, κ 2 ) k G = ([n], E 1 ) H = ([n], E 2 ) (G, κ 1 ) (H, κ 2 ) (G, κ 1 ) (H, κ 2 ) π : [n] [n] σ : [k] [k] (u, v) E 1 (π(u), π(v)) E 2 κ 1 ((u, v)) = σ(κ 2 ((π(u), π(v)))) σ (G, κ 1 ) (H, κ 2 ) (3, 3, 3; 16)

23 (3, 3, 3; n) 14 n 17 R(4, 3, 3) P G P P (A G ) A G G l l k + 1 k = 4 k = 5

24 ν ex(n; {C 3, C 4 })

Breaking Symmetries in Graph Representation

Michael Codish Dept. of Computer Science Ben-Gurion Univ. of the Negev Beer-Sheva, Israel Breaking Symmetries in Graph Representation Alice Miller and Patrick Prosser School of Computing Science University