# Graphical degree sequences and realizations

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

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

swap 1 2 swap- 3 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 ).

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

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.

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?

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

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

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

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), 477 480.

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), 477 480. Greedy algorithm to find a realization

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), 477 480. Greedy algorithm to find a realization time complexity O( d i )

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), 477 480. Greedy algorithm to find a realization time complexity O( d i ) based on swaps

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), 477 480. Greedy algorithm to find a realization time complexity O( d i ) based on swaps blacks G, reds (or blues) are missing

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), 477 480. 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

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), 477 480. 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

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), 477 480. Greedy algorithm to find a realization time complexity O( d i ) based on swaps Let the lexicographic order on [n] [n]

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), 477 480. 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

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), 477 480. 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.

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), 477 480. 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.

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), 477 480. 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

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), 496 506.)

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), 496 506.) from that time on it is called Havel-Hakimi algorithm

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), 496 506.) from that time on it is called Havel-Hakimi algorithm J. K. Senior: Partitions and their Representative Graphs, Amer. J. Math., 73 (1951), 663 689.

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), 496 506.) from that time on it is called Havel-Hakimi algorithm J. K. Senior: Partitions and their Representative Graphs, Amer. J. Math., 73 (1951), 663 689. all possible graphs with multiple edges but no loops to find all possible molecules with given composition

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), 496 506.) from that time on it is called Havel-Hakimi algorithm J. K. Senior: Partitions and their Representative Graphs, Amer. J. Math., 73 (1951), 663 689. all possible graphs with multiple edges but no loops to find all possible molecules with given composition introduced swaps (but called transfusion)

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), 496 506.) from that time on it is called Havel-Hakimi algorithm J. K. Senior: Partitions and their Representative Graphs, Amer. J. Math., 73 (1951), 663 689. 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), 264 274.)

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), 496 506.) from that time on it is called Havel-Hakimi algorithm J. K. Senior: Partitions and their Representative Graphs, Amer. J. Math., 73 (1951), 663 689. 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), 264 274.) used Havel s theorem in the proof

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

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

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

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

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

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 )

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, 1891 - 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 )

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

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), 1073 1082. H.J. Ryser Combinatorial properties of matrices of zeros and ones, Canad. J. Math. 9 (1957), 371 377.

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), 1073 1082. flow theory H.J. Ryser Combinatorial properties of matrices of zeros and ones, Canad. J. Math. 9 (1957), 371 377. binary matrices

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), 1073 1082. flow theory H.J. Ryser Combinatorial properties of matrices of zeros and ones, Canad. J. Math. 9 (1957), 371 377. binary matrices for both it was byproduct to prove EG-type results for directed graphs: no multiply edges, but possible loops

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), 1073 1082. flow theory H.J. Ryser Combinatorial properties of matrices of zeros and ones, Canad. J. Math. 9 (1957), 371 377. 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

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), 1073 1082. flow theory H.J. Ryser Combinatorial properties of matrices of zeros and ones, Canad. J. Math. 9 (1957), 371 377. 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

swap 1 2 swap- 3 4

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

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

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

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.

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

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

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

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).

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.

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

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

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

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

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

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

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

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

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

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.

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

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

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

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

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

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

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

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

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

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

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 ).

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

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

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

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

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

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

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 -

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

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

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 1 4 ) 2 3n ( ) (1 min(d i, V d i ) i ( i ) ( d i 1 4 ) 3n 2 2 3n )

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

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.

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

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

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.

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.

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.

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.

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.

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.

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

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.

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 ).

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

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

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

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 )

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 )

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

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

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

swap 1 2 swap- 3 4

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

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

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

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

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

swap 1 2 swap- 3 4

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

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.

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

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

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

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

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)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 ) )

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

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

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

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

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.

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;

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

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

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.

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. Theorem Each directed sequence realization can be transformed into another one with C 4 - and triangular C 6 -swaps

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. 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

Catherine Greenhill s result swap A polynomial bound on the mixing time of a Markov chain for sampling regular directed graphs, arxiv 1105.0457v4 (2011), 1 48.

Catherine Greenhill s result swap A polynomial bound on the mixing time of a Markov chain for sampling regular directed graphs, arxiv 1105.0457v4 (2011), 1 48. Theorem for regular DD C 4 -swaps only are sufficient

Catherine Greenhill s result swap A polynomial bound on the mixing time of a Markov chain for sampling regular directed graphs, arxiv 1105.0457v4 (2011), 1 48. Theorem for regular DD C 4 -swaps only are sufficient a g c a g c d b e d b e

Catherine Greenhill s result swap A polynomial bound on the mixing time of a Markov chain for sampling regular directed graphs, arxiv 1105.0457v4 (2011), 1 48. 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

Catherine Greenhill s result swap A polynomial bound on the mixing time of a Markov chain for sampling regular directed graphs, arxiv 1105.0457v4 (2011), 1 48. 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