Graphical degree sequences and Definitions realizations and - PowerPoint PPT Presentation

Degree sequences 2a For complete graphs there are much simpler solutions: Swap- distances - Havel A remark on the existence of finite graphs. (Czech), P .L. Erdös est. Mat. 80 (1955), 477–480. ˇ Casopis Pˇ Definitions Greedy algorithm to find a realization and History time complexity O ( � d i ) based on swaps Undirected swap- sequences Let � the lexicographic order on [ n ] × [ n ] Bipartite degree Then � implies lexicographic order on V s.t. sequences [ n ] ↑ = degrees, [ n ] ↓ = subscripts Directed degree - N G ( v ) denotes the neighbors of v in realization G then sequences 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 sequences 2b Swap- distances Hakimi rediscovered ( On the realizability of a set of integers as degrees of P .L. Erdös the vertices of a simple graph. J. SIAM Appl. Math. 10 (1962), 496–506. ) Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Degree sequences 2b Swap- distances Hakimi rediscovered ( On the realizability of a set of integers as degrees of P .L. Erdös the vertices of a simple graph. J. SIAM Appl. Math. 10 (1962), 496–506. ) Definitions and History from that time on it is called Havel-Hakimi algorithm Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Degree sequences 2b Swap- distances Hakimi rediscovered ( On the realizability of a set of integers as degrees of P .L. Erdös the vertices of a simple graph. J. SIAM Appl. Math. 10 (1962), 496–506. ) Definitions and History from that time on it is called Havel-Hakimi algorithm Undirected swap- sequences J. K. Senior: Partitions and their Representative Graphs, Amer. J. Math. , 73 Bipartite (1951), 663–689. degree sequences Directed degree sequences

Degree sequences 2b Swap- distances Hakimi rediscovered ( On the realizability of a set of integers as degrees of P .L. Erdös the vertices of a simple graph. J. SIAM Appl. Math. 10 (1962), 496–506. ) Definitions and History from that time on it is called Havel-Hakimi algorithm Undirected swap- sequences J. K. Senior: Partitions and their Representative Graphs, Amer. J. Math. , 73 Bipartite (1951), 663–689. degree sequences all possible graphs with multiple edges but no loops Directed to find all possible molecules with given composition degree sequences

Degree sequences 2b Swap- distances Hakimi rediscovered ( On the realizability of a set of integers as degrees of P .L. Erdös the vertices of a simple graph. J. SIAM Appl. Math. 10 (1962), 496–506. ) Definitions and History from that time on it is called Havel-Hakimi algorithm Undirected swap- sequences J. K. Senior: Partitions and their Representative Graphs, Amer. J. Math. , 73 Bipartite (1951), 663–689. degree sequences all possible graphs with multiple edges but no loops Directed to find all possible molecules with given composition degree sequences introduced swaps (but called transfusion)

Degree sequences 2b Swap- distances Hakimi rediscovered ( On the realizability of a set of integers as degrees of P .L. Erdös the vertices of a simple graph. J. SIAM Appl. Math. 10 (1962), 496–506. ) Definitions and History from that time on it is called Havel-Hakimi algorithm Undirected swap- sequences J. K. Senior: Partitions and their Representative Graphs, Amer. J. Math. , 73 Bipartite (1951), 663–689. degree sequences all possible graphs with multiple edges but no loops Directed to find all possible molecules with given composition degree sequences introduced swaps (but called transfusion) another method: Erd˝ os-Gallai theorem ( Graphs with prescribed degree of vertices (in Hungarian), Mat. Lapok 11 (1960), 264–274. )

Degree sequences 2b Swap- distances Hakimi rediscovered ( On the realizability of a set of integers as degrees of P .L. Erdös the vertices of a simple graph. J. SIAM Appl. Math. 10 (1962), 496–506. ) Definitions and History from that time on it is called Havel-Hakimi algorithm Undirected swap- sequences J. K. Senior: Partitions and their Representative Graphs, Amer. J. Math. , 73 Bipartite (1951), 663–689. degree sequences all possible graphs with multiple edges but no loops Directed to find all possible molecules with given composition degree sequences introduced swaps (but called transfusion) another method: Erd˝ os-Gallai theorem ( Graphs with prescribed degree of vertices (in Hungarian), Mat. Lapok 11 (1960), 264–274. ) used Havel’s theorem in the proof

Transforming one realization into an other one Swap- Let d graphical degree sequence, G and G ′ two realizations distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Transforming one realization into an other one Swap- Let d graphical degree sequence, G and G ′ two realizations distances P .L. Erdös looking for a sequence of realizations G 1 = H 0 , H 1 , . . . , H k = G 2 Definitions and History s.t. ∀ i = 0 , . . . , k − 1 ∃ swap operation H i → H i + 1 Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Transforming one realization into an other one Swap- Let d graphical degree sequence, G and G ′ two realizations distances P .L. Erdös looking for a sequence of realizations G 1 = H 0 , H 1 , . . . , H k = G 2 Definitions and History s.t. ∀ i = 0 , . . . , k − 1 ∃ swap operation H i → H i + 1 Undirected swap- sequences Lemma Bipartite degree ∃ swap H i → H i + 1 then ∃ swap H i + 1 → H i sequences Directed degree sequences

Transforming one realization into an other one Swap- Let d graphical degree sequence, G and G ′ two realizations distances P .L. Erdös looking for a sequence of realizations G 1 = H 0 , H 1 , . . . , H k = G 2 Definitions and History s.t. ∀ i = 0 , . . . , k − 1 ∃ swap operation H i → H i + 1 Undirected swap- sequences Lemma Bipartite degree ∃ swap H i → H i + 1 then ∃ swap H i + 1 → H i sequences Directed degree sequences by Havel-Hakimi’s lemma such swap-sequence always exists

Transforming one realization into an other one Swap- Let d graphical degree sequence, G and G ′ two realizations distances P .L. Erdös looking for a sequence of realizations G 1 = H 0 , H 1 , . . . , H k = G 2 Definitions and History s.t. ∀ i = 0 , . . . , k − 1 ∃ swap operation H i → H i + 1 Undirected swap- sequences Lemma Bipartite degree ∃ swap H i → H i + 1 then ∃ swap H i + 1 → H i sequences Directed degree sequences 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 degree sequence, G and G ′ two realizations distances P .L. Erdös looking for a sequence of realizations G 1 = H 0 , H 1 , . . . , H k = G 2 Definitions and History s.t. ∀ i = 0 , . . . , k − 1 ∃ swap operation H i → H i + 1 Undirected swap- sequences Lemma Bipartite degree ∃ swap H i → H i + 1 then ∃ swap H i + 1 → H i sequences Directed degree sequences 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 degree sequence, G and G ′ two realizations distances P .L. Erdös looking for a sequence of realizations G 1 = H 0 , H 1 , . . . , H k = G 2 Definitions and History s.t. ∀ i = 0 , . . . , k − 1 ∃ swap operation H i → H i + 1 Undirected swap- sequences Lemma Bipartite degree ∃ swap H i → H i + 1 then ∃ swap H i + 1 → H i sequences Directed degree Theorem (Petersen, 1891 - see Erd˝ os-Gallai paper) sequences by Havel-Hakimi’s lemma such swap-sequence always exists � � for i = 1 , 2 ∃ G i → canonical realizations swap-distance ≤ O ( � d i )

Bipartite and directed cases Swap- distances P .L. Erdös Erd˝ os-Gallai type result for bipartite graphs Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Bipartite and directed cases Swap- distances P .L. Erdös Erd˝ os-Gallai type result for bipartite graphs D. Gale A theorem on flows in networks, Definitions and History Pacific J. Math. 7 (2) (1957), 1073–1082. Undirected H.J. Ryser Combinatorial properties of matrices of zeros and ones, swap- sequences Canad. J. Math. 9 (1957), 371–377. Bipartite degree sequences Directed degree sequences

Bipartite and directed cases Swap- distances P .L. Erdös Erd˝ os-Gallai type result for bipartite graphs D. Gale A theorem on flows in networks, Definitions and History Pacific J. Math. 7 (2) (1957), 1073–1082. flow theory Undirected H.J. Ryser Combinatorial properties of matrices of zeros and ones, swap- sequences Canad. J. Math. 9 (1957), 371–377. binary matrices Bipartite degree sequences Directed degree sequences

Bipartite and directed cases Swap- distances P .L. Erdös Erd˝ os-Gallai type result for bipartite graphs D. Gale A theorem on flows in networks, Definitions and History Pacific J. Math. 7 (2) (1957), 1073–1082. flow theory Undirected H.J. Ryser Combinatorial properties of matrices of zeros and ones, swap- sequences Canad. J. Math. 9 (1957), 371–377. binary matrices Bipartite for both it was byproduct to prove EG-type results for degree sequences directed graphs: no multiply edges, but possible loops Directed degree sequences

Bipartite and directed cases Swap- distances P .L. Erdös Erd˝ os-Gallai type result for bipartite graphs D. Gale A theorem on flows in networks, Definitions and History Pacific J. Math. 7 (2) (1957), 1073–1082. flow theory Undirected H.J. Ryser Combinatorial properties of matrices of zeros and ones, swap- sequences Canad. J. Math. 9 (1957), 371–377. binary matrices Bipartite for both it was byproduct to prove EG-type results for degree sequences directed graphs: no multiply edges, but possible loops Directed degree sequences Both used bipartite graph representation of directed graphs

Bipartite and directed cases Swap- distances P .L. Erdös Erd˝ os-Gallai type result for bipartite graphs D. Gale A theorem on flows in networks, Definitions and History Pacific J. Math. 7 (2) (1957), 1073–1082. flow theory Undirected H.J. Ryser Combinatorial properties of matrices of zeros and ones, swap- sequences Canad. J. Math. 9 (1957), 371–377. binary matrices Bipartite for both it was byproduct to prove EG-type results for degree sequences directed graphs: no multiply edges, but possible loops Directed degree sequences Both used bipartite graph representation of directed graphs Ryser used swap-sequence transformation from one realization to an other one

Swap- distances P .L. Erdös Definitions Definitions and History 1 and History Undirected swap- sequences 2 Undirected swap-sequences Bipartite degree sequences Directed Bipartite degree sequences 3 degree sequences 4 Directed degree sequences

Red/blue graphs Swap- G simple graph with red/blue edges - r ( v ) / b ( v ) degrees distances G is balanced : ∀ v ∈ V ( G ) r ( v ) = b ( v ) . P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Red/blue graphs Swap- G simple graph with red/blue edges - r ( v ) / b ( v ) degrees distances G is balanced : ∀ v ∈ V ( G ) r ( v ) = b ( v ) . P .L. Erdös trail - no multiple edges circuit - closed trail Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Red/blue graphs Swap- G simple graph with red/blue edges - r ( v ) / b ( v ) degrees distances G is balanced : ∀ v ∈ V ( G ) r ( v ) = b ( v ) . P .L. Erdös trail - no multiple edges circuit - closed trail Definitions and History Lemma Undirected swap- balanced ⇒ E ( G ) decomposed to alternating circuits sequences Bipartite degree sequences Directed degree sequences

Red/blue graphs Swap- G simple graph with red/blue edges - r ( v ) / b ( v ) degrees distances G is balanced : ∀ v ∈ V ( G ) r ( v ) = b ( v ) . P .L. Erdös trail - no multiple edges circuit - closed trail Definitions and History Lemma Undirected swap- balanced ⇒ E ( G ) decomposed to alternating circuits sequences Bipartite degree Lemma sequences Directed C = v 1 , v 2 , . . . v 2 n alternating; v i = v j with j − i is even. degree sequences C can be decomposed into two, shorter alternating circuits.

Red/blue graphs Swap- G simple graph with red/blue edges - r ( v ) / b ( v ) degrees distances G is balanced : ∀ v ∈ V ( G ) r ( v ) = b ( v ) . P .L. Erdös trail - no multiple edges circuit - closed trail Definitions and History Lemma Undirected swap- balanced ⇒ E ( G ) decomposed to alternating circuits sequences Bipartite degree Lemma sequences Directed C = v 1 , v 2 , . . . v 2 n alternating; v i = v j with j − i is even. degree sequences 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 ) degrees distances G is balanced : ∀ v ∈ V ( G ) r ( v ) = b ( v ) . P .L. Erdös trail - no multiple edges circuit - closed trail Definitions and History Lemma Undirected swap- balanced ⇒ E ( G ) decomposed to alternating circuits sequences Bipartite degree Lemma sequences Directed C = v 1 , v 2 , . . . v 2 n alternating; v i = v j with j − i is even. degree sequences C can be decomposed into two, shorter alternating circuits. v v v

Red/blue graphs 2 Swap- distances P .L. Erdös maxC u ( G ) = # of circuits in a max. circuit decomposition Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Red/blue graphs 2 Swap- distances P .L. Erdös maxC u ( G ) = # of circuits in a max. circuit decomposition Definitions and History circuit C is elementary if Undirected swap- sequences 1 no vertex appears more than twice in C , Bipartite 2 ∃ i , j s.t. v i and v j occur only once in C and they have degree sequences different parity (their distance is odd). Directed degree sequences

Red/blue graphs 2 Swap- distances P .L. Erdös maxC u ( G ) = # of circuits in a max. circuit decomposition Definitions and History circuit C is elementary if Undirected swap- sequences 1 no vertex appears more than twice in C , Bipartite 2 ∃ i , j s.t. v i and v j occur only once in C and they have degree sequences different parity (their distance is odd). Directed degree sequences Lemma Let C 1 , . . . , C ℓ be a max. size circuit decomposition of G . ⇒ each circuit is elementary.

Red/blue graphs 3 Swap- Proof. distances P .L. Erdös (i) no vertex occurs 3 times Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Red/blue graphs 3 Swap- Proof. distances P .L. Erdös (i) no vertex occurs 3 times (ii) when v occurs twice - their distance is odd Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Red/blue graphs 3 Swap- Proof. distances P .L. Erdös (i) no vertex occurs 3 times (ii) when v occurs twice - their distance is odd Definitions and History (iii) ∃ vertex v occurring once - INDIRECT with min. Undirected distance swap- sequences Bipartite degree sequences Directed degree sequences

Red/blue graphs 3 Swap- Proof. distances P .L. Erdös (i) no vertex occurs 3 times (ii) when v occurs twice - their distance is odd Definitions and History (iii) ∃ vertex v occurring once - INDIRECT with min. Undirected distance swap- sequences v Bipartite degree u 1 sequences Directed degree odd sequences u 1 u 2 v

Red/blue graphs 3 Swap- Proof. distances P .L. Erdös (i) no vertex occurs 3 times (ii) when v occurs twice - their distance is odd Definitions and History (iii) ∃ vertex v occurring once - INDIRECT with min. Undirected distance swap- sequences v Bipartite degree u 1 sequences Directed degree even sequences u 1 u 2 v

Red/blue graphs 3 Swap- Proof. distances P .L. Erdös (i) no vertex occurs 3 times (ii) when v occurs twice - their distance is odd Definitions and History (iii) ∃ vertex v occurring once - INDIRECT with min. Undirected distance swap- sequences v Bipartite degree u 1 sequences Directed degree even sequences u 1 u 2 v (iv) by pigeon hole: ∃ ≥ 2 vertices occurring once

Red/blue graphs 3 Swap- Proof. distances P .L. Erdös (i) no vertex occurs 3 times (ii) when v occurs twice - their distance is odd Definitions and History (iii) ∃ vertex v occurring once - INDIRECT with min. Undirected distance swap- sequences v Bipartite degree u 1 sequences Directed degree even sequences u 1 u 2 v (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- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

One alternating circuit Realizations G 1 and G 2 of d , Swap- distances take E 1 ∆ E 2 = E , and the red/blue graph G = ( V , E ) P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

One alternating circuit Realizations G 1 and G 2 of d , Swap- distances take E 1 ∆ E 2 = E , and the red/blue graph G = ( V , E ) P .L. Erdös Theorem Definitions and History if E ( G ) is one alternating elementary circuit C of length 2 ℓ Undirected swap- ⇒ ∃ swap sequence of length ℓ − 1 from G 1 to G 2 . sequences Bipartite degree sequences Directed degree sequences

One alternating circuit Realizations G 1 and G 2 of d , Swap- distances take E 1 ∆ E 2 = E , and the red/blue graph G = ( V , E ) P .L. Erdös Theorem Definitions and History if E ( G ) is one alternating elementary circuit C of length 2 ℓ Undirected swap- ⇒ ∃ swap sequence of length ℓ − 1 from G 1 to G 2 . sequences Bipartite degree Proof. sequences � � G i start (stop) graphs, induction on actual � E start ∆ E stop Directed � degree sequences

One alternating circuit Realizations G 1 and G 2 of d , Swap- distances take E 1 ∆ E 2 = E , and the red/blue graph G = ( V , E ) P .L. Erdös Theorem Definitions and History if E ( G ) is one alternating elementary circuit C of length 2 ℓ Undirected swap- ⇒ ∃ swap sequence of length ℓ − 1 from G 1 to G 2 . sequences Bipartite degree Proof. sequences � � G i start (stop) graphs, induction on actual � E start ∆ E stop Directed � degree ∃ v ∈ C occurring once, starting a red edge sequences

One alternating circuit Realizations G 1 and G 2 of d , Swap- distances take E 1 ∆ E 2 = E , and the red/blue graph G = ( V , E ) P .L. Erdös Theorem Definitions and History if E ( G ) is one alternating elementary circuit C of length 2 ℓ Undirected swap- ⇒ ∃ swap sequence of length ℓ − 1 from G 1 to G 2 . sequences Bipartite degree Proof. sequences � � G i start (stop) graphs, induction on actual � E start ∆ E stop Directed � degree ∃ v ∈ C occurring once, starting a red edge sequences u red edges miss stop graph v

One alternating circuit Realizations G 1 and G 2 of d , Swap- distances take E 1 ∆ E 2 = E , and the red/blue graph G = ( V , E ) P .L. Erdös Theorem Definitions and History if E ( G ) is one alternating elementary circuit C of length 2 ℓ Undirected swap- ⇒ ∃ swap sequence of length ℓ − 1 from G 1 to G 2 . sequences Bipartite degree Proof. sequences � � G i start (stop) graphs, induction on actual � E start ∆ E stop Directed � degree ∃ v ∈ C occurring once, starting a red edge sequences u red edges miss stop graph if ( u , v ) �∈ E 1 , E 2 v

One alternating circuit Realizations G 1 and G 2 of d , Swap- distances take E 1 ∆ E 2 = E , and the red/blue graph G = ( V , E ) P .L. Erdös Theorem Definitions and History if E ( G ) is one alternating elementary circuit C of length 2 ℓ Undirected swap- ⇒ ∃ swap sequence of length ℓ − 1 from G 1 to G 2 . sequences Bipartite degree Proof. sequences � � G i start (stop) graphs, induction on actual � E start ∆ E stop Directed � degree ∃ v ∈ C occurring once, starting a red edge sequences u red edges miss stop graph if ( u , v ) �∈ E 1 , E 2 v

One alternating circuit Realizations G 1 and G 2 of d , Swap- distances take E 1 ∆ E 2 = E , and the red/blue graph G = ( V , E ) P .L. Erdös Theorem Definitions and History if E ( G ) is one alternating elementary circuit C of length 2 ℓ Undirected swap- ⇒ ∃ swap sequence of length ℓ − 1 from G 1 to G 2 . sequences Bipartite degree Proof. sequences � � G i start (stop) graphs, induction on actual � E start ∆ E stop Directed � degree ∃ v ∈ C occurring once, starting a red edge sequences 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. diff. having 2 edges less v

One alternating circuit Realizations G 1 and G 2 of d , Swap- distances take E 1 ∆ E 2 = E , and the red/blue graph G = ( V , E ) P .L. Erdös Theorem Definitions and History if E ( G ) is one alternating elementary circuit C of length 2 ℓ Undirected swap- ⇒ ∃ swap sequence of length ℓ − 1 from G 1 to G 2 . sequences Bipartite degree Proof. sequences � � G i start (stop) graphs, induction on actual � E start ∆ E stop Directed � degree ∃ v ∈ C occurring once, starting a red edge sequences u red edges miss stop graph if ( u , v ) ∈ E 1 , E 2 v

One alternating circuit Realizations G 1 and G 2 of d , Swap- distances take E 1 ∆ E 2 = E , and the red/blue graph G = ( V , E ) P .L. Erdös Theorem Definitions and History if E ( G ) is one alternating elementary circuit C of length 2 ℓ Undirected swap- ⇒ ∃ swap sequence of length ℓ − 1 from G 1 to G 2 . sequences Bipartite degree Proof. sequences � � G i start (stop) graphs, induction on actual � E start ∆ E stop Directed � degree ∃ v ∈ C occurring once, starting a red edge sequences u red edges miss stop graph if ( u , v ) ∈ E 1 , E 2 one swap in stop graph v

One alternating circuit Realizations G 1 and G 2 of d , Swap- distances take E 1 ∆ E 2 = E , and the red/blue graph G = ( V , E ) P .L. Erdös Theorem Definitions and History if E ( G ) is one alternating elementary circuit C of length 2 ℓ Undirected swap- ⇒ ∃ swap sequence of length ℓ − 1 from G 1 to G 2 . sequences Bipartite degree Proof. sequences � � G i start (stop) graphs, induction on actual � E start ∆ E stop Directed � degree ∃ v ∈ C occurring once, starting a red edge sequences 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. diff. having 2 edges less v

Shortest swap sequences in undirected case G 1 and G 2 realizations of d . Swap- distances dist u ( G 1 , G 2 ) = length of the shortest swap sequence P .L. Erdös maxC u ( G 1 , G 2 ) = # of circuits in a Definitions max. circuit decomposition of E 1 ∆ E 2 and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Shortest swap sequences in undirected case G 1 and G 2 realizations of d . Swap- distances dist u ( G 1 , G 2 ) = length of the shortest swap sequence P .L. Erdös maxC u ( G 1 , G 2 ) = # of circuits in a Definitions max. circuit decomposition of E 1 ∆ E 2 and History Undirected Theorem (Erd˝ os-Király-Miklós, 2012) swap- sequences For all pairs of realizations G 1 , G 2 we have Bipartite degree sequences dist u ( G 1 , G 2 ) = | E 1 ∆ E 2 | − maxC u ( G 1 , G 2 ) . Directed degree 2 sequences

Shortest swap sequences in undirected case G 1 and G 2 realizations of d . Swap- distances dist u ( G 1 , G 2 ) = length of the shortest swap sequence P .L. Erdös maxC u ( G 1 , G 2 ) = # of circuits in a Definitions max. circuit decomposition of E 1 ∆ E 2 and History Undirected Theorem (Erd˝ os-Király-Miklós, 2012) swap- sequences For all pairs of realizations G 1 , G 2 we have Bipartite degree sequences dist u ( G 1 , G 2 ) = | E 1 ∆ E 2 | − maxC u ( G 1 , G 2 ) . Directed degree 2 sequences Very probably the values are NP-complete to be computed

Shortest swap sequences in undirected case G 1 and G 2 realizations of d . Swap- distances dist u ( G 1 , G 2 ) = length of the shortest swap sequence P .L. Erdös maxC u ( G 1 , G 2 ) = # of circuits in a Definitions max. circuit decomposition of E 1 ∆ E 2 and History Undirected Theorem (Erd˝ os-Király-Miklós, 2012) swap- sequences For all pairs of realizations G 1 , G 2 we have Bipartite degree sequences dist u ( G 1 , G 2 ) = | E 1 ∆ E 2 | − maxC u ( G 1 , G 2 ) . Directed degree 2 sequences Very probably the values are NP-complete to be computed New upper bound: dist u ( G 1 , G 2 ) ≤ | E 1 ∆ E 2 | − 1 2

Why a shortest swap-sequences? How to find a typical realization of a degree sequence? Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Why a shortest swap-sequences? How to find a typical realization of a degree sequence? Swap- distances - large social networks only # of connections known (PC) P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Why a shortest swap-sequences? How to find a typical realization of a degree sequence? Swap- distances - large social networks only # of connections known (PC) P .L. Erdös - online growing network modeling Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Why a shortest swap-sequences? How to find a typical realization of a degree sequence? Swap- distances - large social networks only # of connections known (PC) P .L. Erdös - online growing network modeling Definitions and History huge # of realizations - no way to generate all & choose Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Why a shortest swap-sequences? How to find a typical realization of a degree sequence? Swap- distances - large social networks only # of connections known (PC) P .L. Erdös - online growing network modeling Definitions and History huge # of realizations - no way to generate all & choose Undirected swap- Sampling realizations uniformly - sequences Bipartite degree sequences Directed degree sequences

Why a shortest swap-sequences? How to find a typical realization of a degree sequence? Swap- distances - large social networks only # of connections known (PC) P .L. Erdös - online growing network modeling Definitions and History huge # of realizations - no way to generate all & choose Undirected swap- Sampling realizations uniformly - MCMC methods sequences Bipartite degree sequences Directed degree sequences

Why a shortest swap-sequences? How to find a typical realization of a degree sequence? Swap- distances - large social networks only # of connections known (PC) P .L. Erdös - online growing network modeling Definitions and History huge # of realizations - no way to generate all & choose Undirected swap- Sampling realizations uniformly - MCMC methods sequences to estimate mixing time - need to know distances Bipartite degree sequences Directed degree sequences

Why a shortest swap-sequences? How to find a typical realization of a degree sequence? Swap- distances - large social networks only # of connections known (PC) P .L. Erdös - online growing network modeling Definitions and History huge # of realizations - no way to generate all & choose Undirected swap- Sampling realizations uniformly - MCMC methods sequences to estimate mixing time - need to know distances Bipartite degree sequences | E 1 ∆ E 2 | � 1 − 4 � dist u ( G 1 , G 2 ) Directed ≤ · degree 2 3 n sequences �� � � 1 � 2 − 2 ≤ min ( d i , | V | − d i ) 3 n i �� � � 1 − 4 � ≤ d i 3 n i

Proof of shortest swap sequences length (i) ≤ - take a maximal alternating circuit decomposition Swap- distances C 1 , ..., C maxC u ( G 1 , G 2 ) P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Proof of shortest swap sequences length (i) ≤ - take a maximal alternating circuit decomposition Swap- distances C 1 , ..., C maxC u ( G 1 , G 2 ) P .L. Erdös (ia) and realizations G 1 = H 0 , H 1 , . . . , H k − 1 , H k = G 2 s.t. Definitions ∀ i realizations H i and H i + 1 differ exactly in C i . and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Proof of shortest swap sequences length (i) ≤ - take a maximal alternating circuit decomposition Swap- distances C 1 , ..., C maxC u ( G 1 , G 2 ) P .L. Erdös (ia) and realizations G 1 = H 0 , H 1 , . . . , H k − 1 , H k = G 2 s.t. Definitions ∀ i realizations H i and H i + 1 differ exactly in C i . and History - each circuit is elementary Undirected swap- - for all pairs H i , H i + 1 the previous theorem is applicable sequences Bipartite degree sequences Directed degree sequences

Proof of shortest swap sequences length (i) ≤ - take a maximal alternating circuit decomposition Swap- distances C 1 , ..., C maxC u ( G 1 , G 2 ) P .L. Erdös (ib) assume shortest circuit C 1 is the shortest among all Definitions circuits in all possible minimal circuit decomposition and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Proof of shortest swap sequences length (i) ≤ - take a maximal alternating circuit decomposition Swap- distances C 1 , ..., C maxC u ( G 1 , G 2 ) P .L. Erdös (ib) assume shortest circuit C 1 is the shortest among all Definitions circuits in all possible minimal circuit decomposition and History Undirected Lemma swap- sequences � ∃ edge in any other circuits which divides C 1 into two Bipartite degree odd-long trails. sequences Directed degree sequences

Proof of shortest swap sequences length (i) ≤ - take a maximal alternating circuit decomposition Swap- distances C 1 , ..., C maxC u ( G 1 , G 2 ) P .L. Erdös (ib) assume shortest circuit C 1 is the shortest among all Definitions circuits in all possible minimal circuit decomposition and History Undirected Lemma swap- sequences � ∃ edge in any other circuits which divides C 1 into two Bipartite degree odd-long trails. sequences Directed degree sequences

Proof of shortest swap sequences length (i) ≤ - take a maximal alternating circuit decomposition Swap- distances C 1 , ..., C maxC u ( G 1 , G 2 ) P .L. Erdös (ib) assume shortest circuit C 1 is the shortest among all Definitions circuits in all possible minimal circuit decomposition and History Undirected Lemma swap- sequences � ∃ edge in any other circuits which divides C 1 into two Bipartite degree odd-long trails. sequences Directed degree sequences

Proof of shortest swap sequences length (i) ≤ - take a maximal alternating circuit decomposition Swap- distances C 1 , ..., C maxC u ( G 1 , G 2 ) P .L. Erdös (ib) assume shortest circuit C 1 is the shortest among all Definitions circuits in all possible minimal circuit decomposition and History Undirected Lemma swap- sequences � ∃ edge in any other circuits which divides C 1 into two Bipartite degree odd-long trails. sequences Directed degree sequences

Proof of shortest swap sequences length (i) ≤ - take a maximal alternating circuit decomposition Swap- distances C 1 , ..., C maxC u ( G 1 , G 2 ) P .L. Erdös (ib) assume shortest circuit C 1 is the shortest among all Definitions circuits in all possible minimal circuit decomposition and History Undirected Lemma swap- sequences � ∃ edge in any other circuits which divides C 1 into two Bipartite degree odd-long trails. sequences Directed degree sequences

Proof of shortest swap sequences length (i) ≤ - take a maximal alternating circuit decomposition Swap- distances C 1 , ..., C maxC u ( G 1 , G 2 ) P .L. Erdös (ib) assume shortest circuit C 1 is the shortest among all Definitions circuits in all possible minimal circuit decomposition and History Undirected Lemma swap- sequences � ∃ edge in any other circuits which divides C 1 into two Bipartite degree odd-long trails. sequences Directed degree sequences

Proof of shortest swap sequences length (i) ≤ - take a maximal alternating circuit decomposition Swap- distances C 1 , ..., C maxC u ( G 1 , G 2 ) P .L. Erdös (ib) assume shortest circuit C 1 is the shortest among all Definitions circuits in all possible minimal circuit decomposition and History Undirected Lemma swap- sequences � ∃ edge in any other circuits which divides C 1 into two Bipartite degree odd-long trails. sequences Directed degree sequences # of circuits unchanged, ∃ shorter circuit - contradiction

Proof of shortest swap sequences length (i) ≤ - take a maximal alternating circuit decomposition Swap- distances C 1 , ..., C maxC u ( G 1 , G 2 ) P .L. Erdös (ib) assume shortest circuit C 1 is the shortest among all Definitions circuits in all possible minimal circuit decomposition and History Undirected Lemma swap- sequences � ∃ edge in any other circuits which divides C 1 into two Bipartite degree odd-long trails. sequences Directed degree 1 consider the (actual) symmetric difference, sequences 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 sequences length 2 Swap- distances (ii) LHS ≥ RHS - we realign the inequality: P .L. Erdös maxC u ( G 1 , G 2 ) ≥ | E 1 ∆ E 2 | − dist u ( G 1 , G 2 ) . Definitions and History 2 Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Proof of shortest swap sequences length 2 Swap- distances (ii) LHS ≥ RHS - we realign the inequality: P .L. Erdös maxC u ( G 1 , G 2 ) ≥ | E 1 ∆ E 2 | − dist u ( G 1 , G 2 ) . Definitions and History 2 Undirected swap- G 1 = H 0 , H 1 , . . . , H k − 1 , H k = G 2 minimum real. sequence sequences ∀ i the graphs H i and H i + 1 are in swap-distance 1 Bipartite degree sequences Directed degree sequences

Proof of shortest swap sequences length 2 Swap- distances (ii) LHS ≥ RHS - we realign the inequality: P .L. Erdös maxC u ( G 1 , G 2 ) ≥ | E 1 ∆ E 2 | − dist u ( G 1 , G 2 ) . Definitions and History 2 Undirected swap- G 1 = H 0 , H 1 , . . . , H k − 1 , H k = G 2 minimum real. sequence sequences ∀ i the graphs H i and H i + 1 are in swap-distance 1 Bipartite degree swap subsequence from H i to H j also a minimum one sequences Directed degree sequences

Proof of shortest swap sequences length 2 Swap- distances (ii) LHS ≥ RHS - we realign the inequality: P .L. Erdös maxC u ( G 1 , G 2 ) ≥ | E 1 ∆ E 2 | − dist u ( G 1 , G 2 ) . Definitions and History 2 Undirected swap- G 1 = H 0 , H 1 , . . . , H k − 1 , H k = G 2 minimum real. sequence sequences ∀ i the graphs H i and H i + 1 are in swap-distance 1 Bipartite degree swap subsequence from H i to H j also a minimum one sequences Directed degree induction on i sequences

Proof of shortest swap sequences length 2 Swap- distances (ii) LHS ≥ RHS - we realign the inequality: P .L. Erdös maxC u ( G 1 , G 2 ) ≥ | E 1 ∆ E 2 | − dist u ( G 1 , G 2 ) . Definitions and History 2 Undirected swap- G 1 = H 0 , H 1 , . . . , H k − 1 , H k = G 2 minimum real. sequence sequences ∀ i the graphs H i and H i + 1 are in swap-distance 1 Bipartite degree swap subsequence from H i to H j also a minimum one sequences Directed degree induction on i - find circuit decomposition with i circuits: sequences maxC u ( G 1 , H i ) ≥ | E 1 ∆ E ( H i ) | − dist u ( G 1 , H i ) 2