# Graphical degree sequences and realizations

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1 swap Graphical and realizations Péter L. Erdös Alfréd Rényi Institute of Mathematics Hungarian Academy of Sciences MAPCON 12 MPIPKS - Dresden, May 15, 2012

2 swap Graphical and realizations Péter L. Erdös Alfréd Rényi Institute of Mathematics Hungarian Academy of Sciences Joint work with Zoltán Király and István Miklós MAPCON 12 MPIPKS - Dresden, May 15, 2012

3 swap 1 2 swap- 3 4

4 Degree swap G(V; E) simple graph; V = {v 1, v 2,...,v n } nodes positive integers d = (d 1, d 2,..., d n ).

5 Degree swap G(V; E) simple graph; V = {v 1, v 2,...,v n } nodes positive integers d = (d 1, d 2,..., d n ). If simple graph G(V, E) with d(g) = d

6 Degree swap G(V; E) simple graph; V = {v 1, v 2,...,v n } nodes positive integers d = (d 1, d 2,..., d n ). If simple graph G(V, E) with d(g) = d d is a graphical sequence G realizes d.

7 Degree swap G(V; E) simple graph; V = {v 1, v 2,...,v n } nodes positive integers d = (d 1, d 2,..., d n ). If simple graph G(V, E) with d(g) = d d is a graphical sequence G realizes d. Question: how to decide whether d is graphical?

8 Degree swap G(V; E) simple graph; V = {v 1, v 2,...,v n } nodes positive integers d = (d 1, d 2,..., d n ). If simple graph G(V, E) with d(g) = d d is a graphical sequence G realizes d. Question: how to decide whether d is graphical? - Tutte s f -factor theorem (1952) - applied for K n

9 Degree swap G(V; E) simple graph; V = {v 1, v 2,...,v n } nodes positive integers d = (d 1, d 2,..., d n ). If simple graph G(V, E) with d(g) = d d is a graphical sequence G realizes d. Question: how to decide whether d is graphical? - Tutte s f -factor theorem (1952) - applied for K n polynomial algorithm to decide

10 Degree 2a For complete graphs there are much simpler solutions: swap

11 Degree 2a swap For complete graphs there are much simpler solutions: - Havel A remark on the existence of finite graphs. (Czech), Časopis Pěst. Mat. 80 (1955),

12 Degree 2a swap For complete graphs there are much simpler solutions: - Havel A remark on the existence of finite graphs. (Czech), Časopis Pěst. Mat. 80 (1955), Greedy algorithm to find a realization

13 Degree 2a swap For complete graphs there are much simpler solutions: - Havel A remark on the existence of finite graphs. (Czech), Časopis Pěst. Mat. 80 (1955), Greedy algorithm to find a realization time complexity O( d i )

14 Degree 2a swap For complete graphs there are much simpler solutions: - Havel A remark on the existence of finite graphs. (Czech), Časopis Pěst. Mat. 80 (1955), Greedy algorithm to find a realization time complexity O( d i ) based on swaps

15 Degree 2a swap For complete graphs there are much simpler solutions: - Havel A remark on the existence of finite graphs. (Czech), Časopis Pěst. Mat. 80 (1955), Greedy algorithm to find a realization time complexity O( d i ) based on swaps blacks G, reds (or blues) are missing

16 Degree 2a swap For complete graphs there are much simpler solutions: - Havel A remark on the existence of finite graphs. (Czech), Časopis Pěst. Mat. 80 (1955), Greedy algorithm to find a realization time complexity O( d i ) based on swaps blacks G, reds (or blues) are missing blacks are missing reds G

17 Degree 2a swap For complete graphs there are much simpler solutions: - Havel A remark on the existence of finite graphs. (Czech), Časopis Pěst. Mat. 80 (1955), Greedy algorithm to find a realization time complexity O( d i ) based on swaps blacks G, reds (or blues) are missing blacks are missing reds G The new realization satisfies the same sequence

18 Degree 2a swap For complete graphs there are much simpler solutions: - Havel A remark on the existence of finite graphs. (Czech), Časopis Pěst. Mat. 80 (1955), Greedy algorithm to find a realization time complexity O( d i ) based on swaps Let the lexicographic order on [n] [n]

19 Degree 2a swap For complete graphs there are much simpler solutions: - Havel A remark on the existence of finite graphs. (Czech), Časopis Pěst. Mat. 80 (1955), Greedy algorithm to find a realization time complexity O( d i ) based on swaps Let the lexicographic order on [n] [n] Then implies lexicographic order on V s.t. [n] = s, [n] = subscripts

20 Degree 2a swap For complete graphs there are much simpler solutions: - Havel A remark on the existence of finite graphs. (Czech), Časopis Pěst. Mat. 80 (1955), Greedy algorithm to find a realization time complexity O( d i ) based on swaps Let the lexicographic order on [n] [n] Then implies lexicographic order on V s.t. [n] = s, [n] = subscripts - N G (v) denotes the neighbors of v in realization G then Theorem (Havel s Lemma, 1955) If H V \{v} and H = N G (v) and N G (v) H then there exists realization G such that N G (v) = H.

21 Degree 2a swap For complete graphs there are much simpler solutions: - Havel A remark on the existence of finite graphs. (Czech), Časopis Pěst. Mat. 80 (1955), Greedy algorithm to find a realization time complexity O( d i ) based on swaps Let the lexicographic order on [n] [n] Then implies lexicographic order on V s.t. [n] = s, [n] = subscripts - N G (v) denotes the neighbors of v in realization G then Theorem (Havel s Lemma, 1955) If H V \{v} and H = N G (v) and N G (v) H then there exists realization G such that N G (v) = H.

22 Degree 2a swap For complete graphs there are much simpler solutions: - Havel A remark on the existence of finite graphs. (Czech), Časopis Pěst. Mat. 80 (1955), Greedy algorithm to find a realization time complexity O( d i ) based on swaps Let the lexicographic order on [n] [n] Then implies lexicographic order on V s.t. [n] = s, [n] = subscripts - N G (v) denotes the neighbors of v in realization G then Theorem (Havel s Lemma, 1955) If H V \{v} and H = N G (v) and N G (v) H then there exists realization G such that N G (v) = H. there exists canonical realization

23 Degree 2b swap Hakimi rediscovered (On the realizability of a set of integers as s of the vertices of a simple graph. J. SIAM Appl. Math. 10 (1962), )

24 Degree 2b swap Hakimi rediscovered (On the realizability of a set of integers as s of the vertices of a simple graph. J. SIAM Appl. Math. 10 (1962), ) from that time on it is called Havel-Hakimi algorithm

25 Degree 2b swap Hakimi rediscovered (On the realizability of a set of integers as s of the vertices of a simple graph. J. SIAM Appl. Math. 10 (1962), ) from that time on it is called Havel-Hakimi algorithm J. K. Senior: Partitions and their Representative Graphs, Amer. J. Math., 73 (1951),

26 Degree 2b swap Hakimi rediscovered (On the realizability of a set of integers as s of the vertices of a simple graph. J. SIAM Appl. Math. 10 (1962), ) from that time on it is called Havel-Hakimi algorithm J. K. Senior: Partitions and their Representative Graphs, Amer. J. Math., 73 (1951), all possible graphs with multiple edges but no loops to find all possible molecules with given composition

27 Degree 2b swap Hakimi rediscovered (On the realizability of a set of integers as s of the vertices of a simple graph. J. SIAM Appl. Math. 10 (1962), ) from that time on it is called Havel-Hakimi algorithm J. K. Senior: Partitions and their Representative Graphs, Amer. J. Math., 73 (1951), all possible graphs with multiple edges but no loops to find all possible molecules with given composition introduced swaps (but called transfusion)

28 Degree 2b swap Hakimi rediscovered (On the realizability of a set of integers as s of the vertices of a simple graph. J. SIAM Appl. Math. 10 (1962), ) from that time on it is called Havel-Hakimi algorithm J. K. Senior: Partitions and their Representative Graphs, Amer. J. Math., 73 (1951), all possible graphs with multiple edges but no loops to find all possible molecules with given composition introduced swaps (but called transfusion) another method: Erdős-Gallai theorem (Graphs with prescribed of vertices (in Hungarian), Mat. Lapok 11 (1960), )

29 Degree 2b swap Hakimi rediscovered (On the realizability of a set of integers as s of the vertices of a simple graph. J. SIAM Appl. Math. 10 (1962), ) from that time on it is called Havel-Hakimi algorithm J. K. Senior: Partitions and their Representative Graphs, Amer. J. Math., 73 (1951), all possible graphs with multiple edges but no loops to find all possible molecules with given composition introduced swaps (but called transfusion) another method: Erdős-Gallai theorem (Graphs with prescribed of vertices (in Hungarian), Mat. Lapok 11 (1960), ) used Havel s theorem in the proof

30 Transforming one realization into an other one Let d graphical sequence, G and G two realizations swap

31 Transforming one realization into an other one swap Let d graphical sequence, G and G two realizations looking for a sequence of realizations G 1 = H 0, H 1,..., H k = G 2 s.t. i = 0,...,k 1 swap operation H i H i+1

32 Transforming one realization into an other one swap Let d graphical sequence, G and G two realizations looking for a sequence of realizations G 1 = H 0, H 1,..., H k = G 2 s.t. i = 0,...,k 1 swap operation H i H i+1 Lemma swap H i H i+1 then swap H i+1 H i

33 Transforming one realization into an other one swap Let d graphical sequence, G and G two realizations looking for a sequence of realizations G 1 = H 0, H 1,..., H k = G 2 s.t. i = 0,...,k 1 swap operation H i H i+1 Lemma swap H i H i+1 then swap H i+1 H i by Havel-Hakimi s lemma such swap-sequence always exists

34 Transforming one realization into an other one swap Let d graphical sequence, G and G two realizations looking for a sequence of realizations G 1 = H 0, H 1,..., H k = G 2 s.t. i = 0,...,k 1 swap operation H i H i+1 Lemma swap H i H i+1 then swap H i+1 H i by Havel-Hakimi s lemma such swap-sequence always exists ( ) for i = 1, 2 G i canonical realizations

35 Transforming one realization into an other one swap Let d graphical sequence, G and G two realizations looking for a sequence of realizations G 1 = H 0, H 1,..., H k = G 2 s.t. i = 0,...,k 1 swap operation H i H i+1 Lemma swap H i H i+1 then swap H i+1 H i by Havel-Hakimi s lemma such swap-sequence always exists ( ) for i = 1, 2 G i canonical realizations swap-distance O( d i )

36 Transforming one realization into an other one swap Let d graphical sequence, G and G two realizations looking for a sequence of realizations G 1 = H 0, H 1,..., H k = G 2 s.t. i = 0,...,k 1 swap operation H i H i+1 Lemma swap H i H i+1 then swap H i+1 H i Theorem (Petersen, see Erdős-Gallai paper) by Havel-Hakimi s lemma such swap-sequence always exists ( ) for i = 1, 2 G i canonical realizations swap-distance O( d i )

37 and directed cases swap Erdős-Gallai type result for bipartite graphs

38 and directed cases swap Erdős-Gallai type result for bipartite graphs D. Gale A theorem on flows in networks, Pacific J. Math. 7 (2) (1957), H.J. Ryser Combinatorial properties of matrices of zeros and ones, Canad. J. Math. 9 (1957),

39 and directed cases swap Erdős-Gallai type result for bipartite graphs D. Gale A theorem on flows in networks, Pacific J. Math. 7 (2) (1957), flow theory H.J. Ryser Combinatorial properties of matrices of zeros and ones, Canad. J. Math. 9 (1957), binary matrices

40 and directed cases swap Erdős-Gallai type result for bipartite graphs D. Gale A theorem on flows in networks, Pacific J. Math. 7 (2) (1957), flow theory H.J. Ryser Combinatorial properties of matrices of zeros and ones, Canad. J. Math. 9 (1957), binary matrices for both it was byproduct to prove EG-type results for directed graphs: no multiply edges, but possible loops

41 and directed cases swap Erdős-Gallai type result for bipartite graphs D. Gale A theorem on flows in networks, Pacific J. Math. 7 (2) (1957), flow theory H.J. Ryser Combinatorial properties of matrices of zeros and ones, Canad. J. Math. 9 (1957), binary matrices for both it was byproduct to prove EG-type results for directed graphs: no multiply edges, but possible loops Both used bipartite graph representation of directed graphs

42 and directed cases swap Erdős-Gallai type result for bipartite graphs D. Gale A theorem on flows in networks, Pacific J. Math. 7 (2) (1957), flow theory H.J. Ryser Combinatorial properties of matrices of zeros and ones, Canad. J. Math. 9 (1957), binary matrices for both it was byproduct to prove EG-type results for directed graphs: no multiply edges, but possible loops Both used bipartite graph representation of directed graphs Ryser used swap-sequence transformation from one realization to an other one

43 swap 1 2 swap- 3 4

44 Red/blue graphs G simple graph with red/blue edges - r(v) / b(v) s G is balanced : v V(G) r(v) = b(v). swap

45 Red/blue graphs swap G simple graph with red/blue edges - r(v) / b(v) s G is balanced : v V(G) r(v) = b(v). trail - no multiple edges circuit - closed trail

46 Red/blue graphs swap G simple graph with red/blue edges - r(v) / b(v) s G is balanced : v V(G) r(v) = b(v). trail - no multiple edges circuit - closed trail Lemma balanced E(G) decomposed to alternating circuits

47 Red/blue graphs swap G simple graph with red/blue edges - r(v) / b(v) s G is balanced : v V(G) r(v) = b(v). trail - no multiple edges circuit - closed trail Lemma balanced E(G) decomposed to alternating circuits Lemma C = v 1, v 2,...v 2n alternating; v i = v j with j i is even. C can be decomposed into two, shorter alternating circuits.

48 Red/blue graphs swap G simple graph with red/blue edges - r(v) / b(v) s G is balanced : v V(G) r(v) = b(v). trail - no multiple edges circuit - closed trail Lemma balanced E(G) decomposed to alternating circuits Lemma C = v 1, v 2,...v 2n alternating; v i = v j with j i is even. C can be decomposed into two, shorter alternating circuits. v v

49 Red/blue graphs swap G simple graph with red/blue edges - r(v) / b(v) s G is balanced : v V(G) r(v) = b(v). trail - no multiple edges circuit - closed trail Lemma balanced E(G) decomposed to alternating circuits Lemma C = v 1, v 2,...v 2n alternating; v i = v j with j i is even. C can be decomposed into two, shorter alternating circuits. v v v

50 Red/blue graphs 2 swap maxc u (G) = # of circuits in a max. circuit decomposition

51 Red/blue graphs 2 swap maxc u (G) = # of circuits in a max. circuit decomposition circuit C is elementary if 1 no vertex appears more than twice in C, 2 i, j s.t. v i and v j occur only once in C and they have different parity (their distance is odd).

52 Red/blue graphs 2 swap maxc u (G) = # of circuits in a max. circuit decomposition circuit C is elementary if 1 no vertex appears more than twice in C, 2 i, j s.t. v i and v j occur only once in C and they have different parity (their distance is odd). Lemma Let C 1,..., C l be a max. size circuit decomposition of G. each circuit is elementary.

53 Red/blue graphs 3 swap Proof. (i) no vertex occurs 3 times

54 Red/blue graphs 3 swap Proof. (i) no vertex occurs 3 times (ii) when v occurs twice - their distance is odd

55 Red/blue graphs 3 swap Proof. (i) no vertex occurs 3 times (ii) when v occurs twice - their distance is odd (iii) vertex v occurring once - INDIRECT with min. distance

56 u 1 u 2 Red/blue graphs 3 swap Proof. (i) no vertex occurs 3 times (ii) when v occurs twice - their distance is odd (iii) vertex v occurring once - INDIRECT with min. distance v u 1 odd v

57 u 1 u 2 Red/blue graphs 3 swap Proof. (i) no vertex occurs 3 times (ii) when v occurs twice - their distance is odd (iii) vertex v occurring once - INDIRECT with min. distance v u 1 even v

58 Red/blue graphs 3 swap Proof. (i) no vertex occurs 3 times (ii) when v occurs twice - their distance is odd (iii) vertex v occurring once - INDIRECT with min. distance v u 1 even v u 1 u 2 (iv) by pigeon hole: 2 vertices occurring once

59 Red/blue graphs 3 swap Proof. (i) no vertex occurs 3 times (ii) when v occurs twice - their distance is odd (iii) vertex v occurring once - INDIRECT with min. distance v u 1 even v u 1 u 2 (iv) by pigeon hole: 2 vertices occurring once (v) by p.h. : u, v occurring once with odd distance

60 One alternating circuit Realizations G 1 and G 2 of d, swap

61 One alternating circuit Realizations G 1 and G 2 of d, take E 1 E 2 = E, and the red/blue graph G = (V, E) swap

62 One alternating circuit swap Realizations G 1 and G 2 of d, take E 1 E 2 = E, and the red/blue graph G = (V, E) Theorem if E(G) is one alternating elementary circuit C of length 2l swap sequence of length l 1 from G 1 to G 2.

63 One alternating circuit swap Realizations G 1 and G 2 of d, take E 1 E 2 = E, and the red/blue graph G = (V, E) Theorem if E(G) is one alternating elementary circuit C of length 2l swap sequence of length l 1 from G 1 to G 2. Proof. G i start (stop) graphs, induction on actual Estart E stop

64 One alternating circuit swap Realizations G 1 and G 2 of d, take E 1 E 2 = E, and the red/blue graph G = (V, E) Theorem if E(G) is one alternating elementary circuit C of length 2l swap sequence of length l 1 from G 1 to G 2. Proof. G i start (stop) graphs, induction on actual Estart E stop v C occurring once, starting a red edge

65 One alternating circuit swap Realizations G 1 and G 2 of d, take E 1 E 2 = E, and the red/blue graph G = (V, E) Theorem if E(G) is one alternating elementary circuit C of length 2l swap sequence of length l 1 from G 1 to G 2. Proof. G i start (stop) graphs, induction on actual Estart E stop v C occurring once, starting a red edge u red edges miss stop graph v

66 One alternating circuit swap Realizations G 1 and G 2 of d, take E 1 E 2 = E, and the red/blue graph G = (V, E) Theorem if E(G) is one alternating elementary circuit C of length 2l swap sequence of length l 1 from G 1 to G 2. Proof. G i start (stop) graphs, induction on actual Estart E stop v C occurring once, starting a red edge u red edges miss stop graph if (u, v) E 1, E 2 v

67 One alternating circuit swap Realizations G 1 and G 2 of d, take E 1 E 2 = E, and the red/blue graph G = (V, E) Theorem if E(G) is one alternating elementary circuit C of length 2l swap sequence of length l 1 from G 1 to G 2. Proof. G i start (stop) graphs, induction on actual Estart E stop v C occurring once, starting a red edge u red edges miss stop graph if (u, v) E 1, E 2 v

68 One alternating circuit swap Realizations G 1 and G 2 of d, take E 1 E 2 = E, and the red/blue graph G = (V, E) Theorem if E(G) is one alternating elementary circuit C of length 2l swap sequence of length l 1 from G 1 to G 2. Proof. G i start (stop) graphs, induction on actual Estart E stop v C occurring once, starting a red edge u red edges miss stop graph if (u, v) E 1, E 2 one swap in start graph stop graph did not change; new start graph, with sym. v diff. having 2 edges less

69 One alternating circuit swap Realizations G 1 and G 2 of d, take E 1 E 2 = E, and the red/blue graph G = (V, E) Theorem if E(G) is one alternating elementary circuit C of length 2l swap sequence of length l 1 from G 1 to G 2. Proof. G i start (stop) graphs, induction on actual Estart E stop v C occurring once, starting a red edge u red edges miss stop graph if (u, v) E 1, E 2 v

70 One alternating circuit swap Realizations G 1 and G 2 of d, take E 1 E 2 = E, and the red/blue graph G = (V, E) Theorem if E(G) is one alternating elementary circuit C of length 2l swap sequence of length l 1 from G 1 to G 2. Proof. G i start (stop) graphs, induction on actual Estart E stop v C occurring once, starting a red edge u red edges miss stop graph if (u, v) E 1, E 2 one swap in stop graph v

71 One alternating circuit swap Realizations G 1 and G 2 of d, take E 1 E 2 = E, and the red/blue graph G = (V, E) Theorem if E(G) is one alternating elementary circuit C of length 2l swap sequence of length l 1 from G 1 to G 2. Proof. G i start (stop) graphs, induction on actual Estart E stop v C occurring once, starting a red edge u red edges miss stop graph if (u, v) E 1, E 2 one swap in stop graph start graph did not change; new stop graph, with sym. v diff. having 2 edges less

72 Shortest swap in undirected case swap G 1 and G 2 realizations of d. dist u (G 1, G 2 ) = length of the shortest swap sequence maxc u (G 1, G 2 ) = # of circuits in a max. circuit decomposition of E 1 E 2

73 Shortest swap in undirected case swap G 1 and G 2 realizations of d. dist u (G 1, G 2 ) = length of the shortest swap sequence maxc u (G 1, G 2 ) = # of circuits in a max. circuit decomposition of E 1 E 2 Theorem (Erdős-Király-Miklós, 2012) For all pairs of realizations G 1, G 2 we have dist u (G 1, G 2 ) = E 1 E 2 2 maxc u (G 1, G 2 ).

74 Shortest swap in undirected case swap G 1 and G 2 realizations of d. dist u (G 1, G 2 ) = length of the shortest swap sequence maxc u (G 1, G 2 ) = # of circuits in a max. circuit decomposition of E 1 E 2 Theorem (Erdős-Király-Miklós, 2012) For all pairs of realizations G 1, G 2 we have dist u (G 1, G 2 ) = E 1 E 2 2 maxc u (G 1, G 2 ). Very probably the values are NP-complete to be computed

75 Shortest swap in undirected case swap G 1 and G 2 realizations of d. dist u (G 1, G 2 ) = length of the shortest swap sequence maxc u (G 1, G 2 ) = # of circuits in a max. circuit decomposition of E 1 E 2 Theorem (Erdős-Király-Miklós, 2012) For all pairs of realizations G 1, G 2 we have dist u (G 1, G 2 ) = E 1 E 2 2 maxc u (G 1, G 2 ). Very probably the values are NP-complete to be computed New upper bound: dist u (G 1, G 2 ) E 1 E 2 2 1

76 Why a shortest swap-? How to find a typical realization of a sequence? swap

77 Why a shortest swap-? How to find a typical realization of a sequence? - large social networks only # of connections known (PC) swap

78 Why a shortest swap-? swap How to find a typical realization of a sequence? - large social networks only # of connections known (PC) - online growing network modeling

79 Why a shortest swap-? swap How to find a typical realization of a sequence? - large social networks only # of connections known (PC) - online growing network modeling huge # of realizations - no way to generate all & choose

80 Why a shortest swap-? swap How to find a typical realization of a sequence? - large social networks only # of connections known (PC) - online growing network modeling huge # of realizations - no way to generate all & choose Sampling realizations uniformly -

81 Why a shortest swap-? swap How to find a typical realization of a sequence? - large social networks only # of connections known (PC) - online growing network modeling huge # of realizations - no way to generate all & choose Sampling realizations uniformly - MCMC methods

82 Why a shortest swap-? swap How to find a typical realization of a sequence? - large social networks only # of connections known (PC) - online growing network modeling huge # of realizations - no way to generate all & choose Sampling realizations uniformly - MCMC methods to estimate mixing time - need to know distances

83 Why a shortest swap-? swap How to find a typical realization of a sequence? - large social networks only # of connections known (PC) - online growing network modeling huge # of realizations - no way to generate all & choose Sampling realizations uniformly - MCMC methods to estimate mixing time - need to know distances dist u (G 1, G 2 ) E ( 1 E ) 2 3n ( ) (1 min(d i, V d i ) i ( i ) ( d i 1 4 ) 3n 2 2 3n )

84 Proof of shortest swap length (i) - take a maximal alternating circuit decomposition C 1,..., C maxcu(g 1,G 2 ) swap

85 Proof of shortest swap length swap (i) - take a maximal alternating circuit decomposition C 1,..., C maxcu(g 1,G 2 ) (ia) and realizations G 1 = H 0, H 1,..., H k 1, H k = G 2 s.t. i realizations H i and H i+1 differ exactly in C i.

86 Proof of shortest swap length swap (i) - take a maximal alternating circuit decomposition C 1,..., C maxcu(g 1,G 2 ) (ia) and realizations G 1 = H 0, H 1,..., H k 1, H k = G 2 s.t. i realizations H i and H i+1 differ exactly in C i. - each circuit is elementary - for all pairs H i, H i+1 the previous theorem is applicable

87 Proof of shortest swap length swap (i) - take a maximal alternating circuit decomposition C 1,..., C maxcu(g 1,G 2 ) (ib) assume shortest circuit C 1 is the shortest among all circuits in all possible minimal circuit decomposition

88 Proof of shortest swap length swap (i) - take a maximal alternating circuit decomposition C 1,..., C maxcu(g 1,G 2 ) (ib) assume shortest circuit C 1 is the shortest among all circuits in all possible minimal circuit decomposition Lemma edge in any other circuits which divides C 1 into two odd-long trails.

89 Proof of shortest swap length swap (i) - take a maximal alternating circuit decomposition C 1,..., C maxcu(g 1,G 2 ) (ib) assume shortest circuit C 1 is the shortest among all circuits in all possible minimal circuit decomposition Lemma edge in any other circuits which divides C 1 into two odd-long trails.

90 Proof of shortest swap length swap (i) - take a maximal alternating circuit decomposition C 1,..., C maxcu(g 1,G 2 ) (ib) assume shortest circuit C 1 is the shortest among all circuits in all possible minimal circuit decomposition Lemma edge in any other circuits which divides C 1 into two odd-long trails.

91 Proof of shortest swap length swap (i) - take a maximal alternating circuit decomposition C 1,..., C maxcu(g 1,G 2 ) (ib) assume shortest circuit C 1 is the shortest among all circuits in all possible minimal circuit decomposition Lemma edge in any other circuits which divides C 1 into two odd-long trails.

92 Proof of shortest swap length swap (i) - take a maximal alternating circuit decomposition C 1,..., C maxcu(g 1,G 2 ) (ib) assume shortest circuit C 1 is the shortest among all circuits in all possible minimal circuit decomposition Lemma edge in any other circuits which divides C 1 into two odd-long trails.

93 Proof of shortest swap length swap (i) - take a maximal alternating circuit decomposition C 1,..., C maxcu(g 1,G 2 ) (ib) assume shortest circuit C 1 is the shortest among all circuits in all possible minimal circuit decomposition Lemma edge in any other circuits which divides C 1 into two odd-long trails.

94 Proof of shortest swap length swap (i) - take a maximal alternating circuit decomposition C 1,..., C maxcu(g 1,G 2 ) (ib) assume shortest circuit C 1 is the shortest among all circuits in all possible minimal circuit decomposition Lemma edge in any other circuits which divides C 1 into two odd-long trails. # of circuits unchanged, shorter circuit - contradiction

95 Proof of shortest swap length swap (i) - take a maximal alternating circuit decomposition C 1,..., C maxcu(g 1,G 2 ) (ib) assume shortest circuit C 1 is the shortest among all circuits in all possible minimal circuit decomposition Lemma edge in any other circuits which divides C 1 into two odd-long trails. 1 consider the (actual) symmetric difference, 2 find a maximal circuit decomposition with a shortest elementary circuit, 3 apply the procedure of one elementary circuit, 4 repeat the whole process with the new (and smaller) symmetric difference.

96 Proof of shortest swap length 2 swap (ii) LHS RHS - we realign the inequality: maxc u (G 1, G 2 ) E 1 E 2 2 dist u (G 1, G 2 ).

97 Proof of shortest swap length 2 swap (ii) LHS RHS - we realign the inequality: maxc u (G 1, G 2 ) E 1 E 2 2 dist u (G 1, G 2 ). G 1 = H 0, H 1,...,H k 1, H k = G 2 minimum real. sequence i the graphs H i and H i+1 are in swap-distance 1

98 Proof of shortest swap length 2 swap (ii) LHS RHS - we realign the inequality: maxc u (G 1, G 2 ) E 1 E 2 2 dist u (G 1, G 2 ). G 1 = H 0, H 1,...,H k 1, H k = G 2 minimum real. sequence i the graphs H i and H i+1 are in swap-distance 1 swap subsequence from H i to H j also a minimum one

99 Proof of shortest swap length 2 swap (ii) LHS RHS - we realign the inequality: maxc u (G 1, G 2 ) E 1 E 2 2 dist u (G 1, G 2 ). G 1 = H 0, H 1,...,H k 1, H k = G 2 minimum real. sequence i the graphs H i and H i+1 are in swap-distance 1 swap subsequence from H i to H j also a minimum one induction on i

100 Proof of shortest swap length 2 swap (ii) LHS RHS - we realign the inequality: maxc u (G 1, G 2 ) E 1 E 2 2 dist u (G 1, G 2 ). G 1 = H 0, H 1,...,H k 1, H k = G 2 minimum real. sequence i the graphs H i and H i+1 are in swap-distance 1 swap subsequence from H i to H j also a minimum one induction on i - find circuit decomposition with i circuits: maxc u (G 1, H i ) E 1 E(H i ) 2 dist u (G 1, H i )

101 Proof of shortest swap length 2 swap (ii) LHS RHS - we realign the inequality: maxc u (G 1, G 2 ) E 1 E 2 2 dist u (G 1, G 2 ). G 1 = H 0, H 1,...,H k 1, H k = G 2 minimum real. sequence i the graphs H i and H i+1 are in swap-distance 1 swap subsequence from H i to H j also a minimum one induction on i - find circuit decomposition with i circuits: maxc u (G 1, H i ) E 1 E(H i ) 2 dist u (G 1, H i ) analyze the intersection of E(H i ) E(H i+1 ) with E 1 E(H i )

102 Proof of shortest swap length 2 swap (ii) LHS RHS - we realign the inequality: maxc u (G 1, G 2 ) E 1 E 2 2 dist u (G 1, G 2 ). G 1 = H 0, H 1,...,H k 1, H k = G 2 minimum real. sequence i the graphs H i and H i+1 are in swap-distance 1 swap subsequence from H i to H j also a minimum one induction on i - find circuit decomposition with i circuits: maxc u (G 1, H i ) E 1 E(H i ) 2 dist u (G 1, H i ) analyze the intersection of E(H i ) E(H i+1 ) with E 1 E(H i ) empty

103 Proof of shortest swap length 2 swap (ii) LHS RHS - we realign the inequality: maxc u (G 1, G 2 ) E 1 E 2 2 dist u (G 1, G 2 ). G 1 = H 0, H 1,...,H k 1, H k = G 2 minimum real. sequence i the graphs H i and H i+1 are in swap-distance 1 swap subsequence from H i to H j also a minimum one induction on i - find circuit decomposition with i circuits: maxc u (G 1, H i ) E 1 E(H i ) 2 dist u (G 1, H i ) analyze the intersection of E(H i ) E(H i+1 ) with E 1 E(H i ) empty not empty

104 Proof of shortest swap length 2 swap (ii) LHS RHS - we realign the inequality: maxc u (G 1, G 2 ) E 1 E 2 2 dist u (G 1, G 2 ). G 1 = H 0, H 1,...,H k 1, H k = G 2 minimum real. sequence i the graphs H i and H i+1 are in swap-distance 1 swap subsequence from H i to H j also a minimum one induction on i - find circuit decomposition with i circuits: maxc u (G 1, H i ) E 1 E(H i ) 2 dist u (G 1, H i ) analyze the intersection of E(H i ) E(H i+1 ) with E 1 E(H i ) empty not empty

105 swap 1 2 swap- 3 4

106 G(U, V; E) simple bipartite graph, bipartite sequence: (l k) bd(g) = ( (a1,..., a k ), ( b1,...,b l ) ), swap

107 swap G(U, V; E) simple bipartite graph, bipartite sequence: (l k) bd(g) = ( (a1,..., a k ), ( b1,...,b l ) ), everything goes through - but be careful - f.e. with swap

108 swap G(U, V; E) simple bipartite graph, bipartite sequence: (l k) bd(g) = ( (a1,..., a k ), ( b1,...,b l ) ), everything goes through - but be careful - f.e. with swap maximum circuit decomposition = set of elementary cycles

109 swap G(U, V; E) simple bipartite graph, bipartite sequence: (l k) bd(g) = ( (a1,..., a k ), ( b1,...,b l ) ), everything goes through - but be careful - f.e. with swap maximum circuit decomposition = set of elementary cycles the cycles can be processed in an arbitrary order

110 swap G(U, V; E) simple bipartite graph, bipartite sequence: (l k) bd(g) = ( (a1,..., a k ), ( b1,...,b l ) ), everything goes through - but be careful - f.e. with swap maximum circuit decomposition = set of elementary cycles the cycles can be processed in an arbitrary order dist u (B 1, B 2 ) E(B 1) E(B 2 ) l 1 ( 2 l 2 min ( ) ) ( 1 a i,l a i 2 1 ) 2l i ( ) l 1 a i. l i

111 swap 1 2 swap- 3 4

112 swap G(X; E) simple directed graph, X = {x 1, x 2,..., x n } dd( G) ( (d + = 1, ) ( d+ 2,...,d+ n, d 1, d 2,..., ) ) d n

113 swap G(X; E) simple directed graph, X = {x 1, x 2,..., x n } dd( G) ( (d + = 1, ) ( d+ 2,...,d+ n, d 1, d 2,..., ) ) d n representative bipartite graph B( G) = (U, V; E) (Gale) u i U - out-edges from v i V in-edges to x i.

114 swap G(X; E) simple directed graph, X = {x 1, x 2,..., x n } dd( G) ( (d + = 1, ) ( d+ 2,...,d+ n, d 1, d 2,..., ) ) d n representative bipartite graph B( G) = (U, V; E) (Gale) u i U - out-edges from v i V in-edges to x i. E ( B( G 1 ) ) E(B ( G2 ) ) v 4 u 3 u 1 v 2

115 swap G(X; E) simple directed graph, X = {x 1, x 2,..., x n } dd( G) ( (d + = 1, ) ( d+ 2,...,d+ n, d 1, d 2,..., ) ) d n representative bipartite graph B( G) = (U, V; E) (Gale) u i U - out-edges from v i V in-edges to x i. E ( B( G 1 ) ) E(B ( G2 ) ) u 3 x 4 v 4 E( G 1 ) E( G 2 ) x 3 u 1 v 2 x 1 x 2

116 Possible problems in E(B 1 ) E(B 2 ) Goal: apply results on bipartite for directed. swap

117 Possible problems in E(B 1 ) E(B 2 ) swap Goal: apply results on bipartite for directed. there are two problems

118 Possible problems in E(B 1 ) E(B 2 ) swap Goal: apply results on bipartite for directed. there are two problems v z u u x v y y, z x where x, y, z V( G)

119 Possible problems in E(B 1 ) E(B 2 ) swap Goal: apply results on bipartite for directed. there are two problems v z u x u v y v z u y u a v x v b u c y, z x where x, y, z V( G) if a = x swap in u a, v b, u c, v x

120 Possible problems in E(B 1 ) E(B 2 ) swap Goal: apply results on bipartite for directed. there are two problems v z u x u v y v z u y u a v x v b u c y, z x where x, y, z V( G) if a = x swap in u a, v b, u c, v x if b y swap in v b, u c, v x, u y

121 Handling elementary circuits(=cycles) (i) u a C s.t. v a C swap

122 Handling elementary circuits(=cycles) (i) u a C s.t. v a C swap u y v x v z u c u a v b

123 Handling elementary circuits(=cycles) (i) u a C s.t. v a C swap u y v x v z u c u a v b if is not an edge

124 Handling elementary circuits(=cycles) (i) u a C s.t. v a C swap u y v x v z u c u a v b

125 Handling elementary circuits(=cycles) (i) u a C s.t. v a C (ii) if x : u x C v x C but x : v x u x 3 swap

126 Handling elementary circuits(=cycles) (i) u a C s.t. v a C (ii) if x : u x C v x C but x : v x u x 3 u b v a v x swap v c u z u x v y if is not an edge

127 Handling elementary circuits(=cycles) (i) u a C s.t. v a C (ii) if x : u x C v x C but x : v x u x 3 u b v a v x swap v c u z u x v y

128 Handling elementary circuits(=cycles) swap (i) u a C s.t. v a C (ii) if x : u x C v x C but x : v x u x 3 (iii) if x : if u i = u x C and v i+1 = v x C but C > 6

129 Handling elementary circuits(=cycles) swap (i) u a C s.t. v a C (ii) if x : u x C v x C but x : v x u x 3 (iii) if x : if u i = u x C and v i+1 = v x C but C > 6 u 5 v 3 u 4 v 2 u 3 v 4 u 1 v 5 u 2 v 1

130 Handling elementary circuits(=cycles) swap (i) u a C s.t. v a C (ii) if x : u x C v x C but x : v x u x 3 (iii) if x : if u i = u x C and v i+1 = v x C but C > 6 u 5 v 3 u 4 v 2 u 3 v 4 u 1 v 5 u 2 v 1

131 Handling elementary circuits(=cycles) swap (i) u a C s.t. v a C (ii) if x : u x C v x C but x : v x u x 3 (iii) if x : if u i = u x C and v i+1 = v x C but C > 6 all these swaps are C 4 -swaps

132 Handling elementary circuits(=cycles) swap (i) u a C s.t. v a C (ii) if x : u x C v x C but x : v x u x 3 (iii) if x : if u i = u x C and v i+1 = v x C but C > 6 all these swaps are C 4 -swaps (iv) if x : if u i = u x C and v i+1 = v x C but C = 6

133 Handling elementary circuits(=cycles) swap (i) u a C s.t. v a C (ii) if x : u x C v x C but x : v x u x 3 (iii) if x : if u i = u x C and v i+1 = v x C but C > 6 all these swaps are C 4 -swaps (iv) if x : if u i = u x C and v i+1 = v x C but C = 6 u y v x u z v z v y u x

134 Handling elementary circuits(=cycles) swap (i) u a C s.t. v a C (ii) if x : u x C v x C but x : v x u x 3 (iii) if x : if u i = u x C and v i+1 = v x C but C > 6 all these swaps are C 4 -swaps (iv) if x : if u i = u x C and v i+1 = v x C but C = 6 u y v x u z v z v y u x

135 Handling elementary circuits(=cycles) swap (i) u a C s.t. v a C (ii) if x : u x C v x C but x : v x u x 3 (iii) if x : if u i = u x C and v i+1 = v x C but C > 6 all these swaps are C 4 -swaps (iv) if x : if u i = u x C and v i+1 = v x C but C = 6 u y v x u z v z v y u x triangular C 6 -swap

136 Handling elementary circuits(=cycles) swap (i) u a C s.t. v a C (ii) if x : u x C v x C but x : v x u x 3 (iii) if x : if u i = u x C and v i+1 = v x C but C > 6 all these swaps are C 4 -swaps (iv) if x : if u i = u x C and v i+1 = v x C but C = 6 u y v x u z v z v y u x triangular C 6 -swap triangular C 6 -swaps can be necessary

137 Analyzing triangular C 6 c e swap b a d G 1 and G 1 on common set of vertices

138 Analyzing triangular C 6 c e swap b a d G 1 and G 1 on common set of vertices

139 Analyzing triangular C 6 c e v e swap a v d u a v c u b u d b d G 1 and G 1 on common set of vertices v b u c v a u e E ( B( G 1 ) ) E ( B( G 1 ) )

140 Analyzing triangular C 6 there are two different alternating cycle decompositions swap

141 Analyzing triangular C 6 swap there are two different alternating cycle decompositions u a v c u b v b u c va u e v a v d u d u a v e

142 Analyzing triangular C 6 there are two different alternating cycle decompositions v e swap u a u c v b u d u a v a u d v b uc v a v e

143 Analyzing triangular C 6 swap Whenever a triangular C 6 kisses another elementary cycle in the decomposition, they can be re-decomposed without triangular C 6

144 Analyzing triangular C 6 swap Whenever a triangular C 6 kisses another elementary cycle in the decomposition, they can be re-decomposed without triangular C 6 Lemma maximum size C of E 1 E 2 having minimum # triangular C 6 Then no triangular C 6 kisses any other cycle.

145 Analyzing triangular C 6 swap Whenever a triangular C 6 kisses another elementary cycle in the decomposition, they can be re-decomposed without triangular C 6 Lemma maximum size C of E 1 E 2 having minimum # triangular C 6 Then no triangular C 6 kisses any other cycle. weighted swap distance weight(c 4 -swap) = 1;

146 Analyzing triangular C 6 swap Whenever a triangular C 6 kisses another elementary cycle in the decomposition, they can be re-decomposed without triangular C 6 Lemma maximum size C of E 1 E 2 having minimum # triangular C 6 Then no triangular C 6 kisses any other cycle. weighted swap distance weight(c 4 -swap) = 1; weight(triangular C 6 -swap) = 2

147 Analyzing triangular C 6 swap Whenever a triangular C 6 kisses another elementary cycle in the decomposition, they can be re-decomposed without triangular C 6 Lemma maximum size C of E 1 E 2 having minimum # triangular C 6 Then no triangular C 6 kisses any other cycle. weighted swap distance weight(c 4 -swap) = 1; weight(triangular C 6 -swap) = 2 Theorem Let dd be a directed sequence with G 1 and G 2 realizations. Then dist d ( G 1, G 2 ) = E 1 E 2 maxc d (G 1, G 2 ). 2

148 M.Drew LaMar s result swap 3-Cycle Anchored Digraphs And Their Application In The Uniform Sampling Of Realizations From A Fixed Degree Sequence, in ACM 2011 Winter Simulation Conference (2011), 1 12.

149 M.Drew LaMar s result swap 3-Cycle Anchored Digraphs And Their Application In The Uniform Sampling Of Realizations From A Fixed Degree Sequence, in ACM 2011 Winter Simulation Conference (2011), Theorem Each directed sequence realization can be transformed into another one with C 4 - and triangular C 6 -swaps

150 M.Drew LaMar s result swap 3-Cycle Anchored Digraphs And Their Application In The Uniform Sampling Of Realizations From A Fixed Degree Sequence, in ACM 2011 Winter Simulation Conference (2011), Theorem Each directed sequence realization can be transformed into another one with C 4 - and triangular C 6 -swaps - allowing all C 6 -swaps with weight 2 we have Theorem dist d ( G 1, G 2 ) can be achieved with C 4 - and triangular C 6 -swaps only

151 Catherine Greenhill s result swap A polynomial bound on the mixing time of a Markov chain for sampling regular directed graphs, arxiv v4 (2011), 1 48.

152 Catherine Greenhill s result swap A polynomial bound on the mixing time of a Markov chain for sampling regular directed graphs, arxiv v4 (2011), Theorem for regular DD C 4 -swaps only are sufficient

153 Catherine Greenhill s result swap A polynomial bound on the mixing time of a Markov chain for sampling regular directed graphs, arxiv v4 (2011), Theorem for regular DD C 4 -swaps only are sufficient a g c a g c d b e d b e

154 Catherine Greenhill s result swap A polynomial bound on the mixing time of a Markov chain for sampling regular directed graphs, arxiv v4 (2011), Theorem for regular DD C 4 -swaps only are sufficient a g c a g c d b e d b e swap sequence generated by Greenhill cannot be a minimal

155 Catherine Greenhill s result swap A polynomial bound on the mixing time of a Markov chain for sampling regular directed graphs, arxiv v4 (2011), Theorem for regular DD C 4 -swaps only are sufficient a g c a g c d b e d b e swap sequence generated by Greenhill cannot be a minimal of course this was never a requirement

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