Graphical degree sequences and realizations


 Garey Rogers
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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 HavelHakimi 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 HavelHakimi 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 HavelHakimi 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 HavelHakimi 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 HavelHakimi 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: ErdősGallai 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 HavelHakimi 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: ErdősGallai 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 HavelHakimi s lemma such swapsequence 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 HavelHakimi s lemma such swapsequence 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 HavelHakimi s lemma such swapsequence always exists ( ) for i = 1, 2 G i canonical realizations swapdistance 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 ErdősGallai paper) by HavelHakimi s lemma such swapsequence always exists ( ) for i = 1, 2 G i canonical realizations swapdistance O( d i )
37 and directed cases swap ErdősGallai type result for bipartite graphs
38 and directed cases swap ErdősGallai 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 ErdősGallai 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 ErdősGallai 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 EGtype results for directed graphs: no multiply edges, but possible loops
41 and directed cases swap ErdősGallai 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 EGtype results for directed graphs: no multiply edges, but possible loops Both used bipartite graph representation of directed graphs
42 and directed cases swap ErdősGallai 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 EGtype results for directed graphs: no multiply edges, but possible loops Both used bipartite graph representation of directed graphs Ryser used swapsequence 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 (ErdősKirályMiklós, 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 (ErdősKirályMiklós, 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 NPcomplete 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 (ErdősKirályMiklós, 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 NPcomplete 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 oddlong 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 oddlong 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 oddlong 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 oddlong 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 oddlong 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 oddlong 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 oddlong 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 oddlong 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 swapdistance 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 swapdistance 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 swapdistance 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 swapdistance 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 swapdistance 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 swapdistance 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 swapdistance 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 swapdistance 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  outedges from v i V inedges 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  outedges from v i V inedges 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  outedges from v i V inedges 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 redecomposed 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 redecomposed 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 redecomposed 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 redecomposed 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 redecomposed 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 3Cycle 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 3Cycle 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 3Cycle 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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