On packing of arborescences with matroid constraints Zolt an - PowerPoint PPT Presentation

Motivation 1 : Undirected = Orientation + Directed Theorem (Tutte, Nash-Williams 1961) Let G be an undirected graph and k a positive integer. There exists a packing of k spanning trees in G ⇐ ⇒ G is k-partition-connected. Theorem (Edmonds 1973) Let D be a directed graph, r a vertex of D and k a positive integer. ⇐ ⇒ There exists a packing of k spanning r-arborescences in D D is k-rooted-connected for r. Theorem (Frank 1978) Let G be an undirected graph, r a vertex of G and k a positive integer. There exists an orientation of G that is k-rooted-connected for r ⇐ ⇒ G is k-partition-connected. Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 3 / 16

Motivation 1 : Undirected = Orientation + Directed Theorem (Tutte, Nash-Williams 1961) Let G be an undirected graph and k a positive integer. There exists a packing of k spanning trees in G ⇐ ⇒ G is k-partition-connected. Theorem (Edmonds 1973) Let D be a directed graph, r a vertex of D and k a positive integer. ⇐ ⇒ There exists a packing of k spanning r-arborescences in D D is k-rooted-connected for r. Theorem (Frank 1978) Let G be an undirected graph, r a vertex of G and k a positive integer. There exists an orientation of G that is k-rooted-connected for r ⇐ ⇒ G is k-partition-connected. Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 3 / 16

Motivation 1 : Undirected = Orientation + Directed Theorem (Tutte, Nash-Williams 1961) Let G be an undirected graph and k a positive integer. There exists a packing of k spanning trees in G ⇐ ⇒ G is k-partition-connected. Theorem (Edmonds 1973) Let D be a directed graph, r a vertex of D and k a positive integer. ⇐ ⇒ There exists a packing of k spanning r-arborescences in D D is k-rooted-connected for r. Theorem (Frank 1978) Let G be an undirected graph, r a vertex of G and k a positive integer. There exists an orientation of G that is k-rooted-connected for r ⇐ ⇒ G is k-partition-connected. Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 3 / 16

Motivation 1 : Undirected = Orientation + Directed Theorem (Tutte, Nash-Williams 1961) Let G be an undirected graph and k a positive integer. There exists a packing of k spanning trees in G ⇐ ⇒ G is k-partition-connected. Theorem (Edmonds 1973) Let D be a directed graph, r a vertex of D and k a positive integer. ⇐ ⇒ There exists a packing of k spanning r-arborescences in D D is k-rooted-connected for r. Theorem (Frank 1978) Let G be an undirected graph, r a vertex of G and k a positive integer. There exists an orientation of G that is k-rooted-connected for r ⇐ ⇒ G is k-partition-connected. Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 3 / 16

Motivation 1 : Undirected = Orientation + Directed Theorem (Tutte, Nash-Williams 1961) Let G be an undirected graph and k a positive integer. There exists a packing of k spanning trees in G ⇐ ⇒ G is k-partition-connected. Theorem (Edmonds 1973) Let D be a directed graph, r a vertex of D and k a positive integer. ⇐ ⇒ There exists a packing of k spanning r-arborescences in D D is k-rooted-connected for r. Theorem (Frank 1978) Let G be an undirected graph, r a vertex of G and k a positive integer. There exists an orientation of G that is k-rooted-connected for r ⇐ ⇒ G is k-partition-connected. Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 3 / 16

Motivation 1 : Undirected = Orientation + Directed Theorem (Tutte, Nash-Williams 1961) Let G be an undirected graph and k a positive integer. There exists a packing of k spanning trees in G ⇐ ⇒ G is k-partition-connected. Theorem (Edmonds 1973) Let D be a directed graph, r a vertex of D and k a positive integer. ⇐ ⇒ There exists a packing of k spanning r-arborescences in D D is k-rooted-connected for r. Theorem (Frank 1978) Let G be an undirected graph, r a vertex of G and k a positive integer. There exists an orientation of G that is k-rooted-connected for r ⇐ ⇒ G is k-partition-connected. Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 3 / 16

Motivation 1 : Undirected = Orientation + Directed Theorem (Tutte, Nash-Williams 1961) Let G be an undirected graph and k a positive integer. There exists a packing of k spanning trees in G ⇐ ⇒ G is k-partition-connected. Theorem (Edmonds 1973) Let D be a directed graph, r a vertex of D and k a positive integer. ⇐ ⇒ There exists a packing of k spanning r-arborescences in D D is k-rooted-connected for r. Theorem (Frank 1978) Let G be an undirected graph, r a vertex of G and k a positive integer. There exists an orientation of G that is k-rooted-connected for r ⇐ ⇒ G is k-partition-connected. Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 3 / 16

Motivation 2 : Rigidity Body-Bar Framework Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 4 / 16

Motivation 2 : Rigidity Body-Bar Framework Theorem (Tay 1984) ”Rigidity” of a Body-Bar Framework can be characterized by the existence of a spanning tree decomposition. Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 4 / 16

Motivation 2 : Rigidity Body-Bar Framework Theorem (Tay 1984) ”Rigidity” of a Body-Bar Framework can be characterized by the existence of a spanning tree decomposition. Body-Bar Framework with Bar-Boundary Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 4 / 16

Motivation 2 : Rigidity Body-Bar Framework Theorem (Tay 1984) ”Rigidity” of a Body-Bar Framework can be characterized by the existence of a spanning tree decomposition. Body-Bar Framework Theorem (Katoh, Tanigawa 2012) with Bar-Boundary ”Rigidity” of a Body-Bar Framework with Bar-Boundary can be characterized by the existence of a matroid-based rooted-tree decomposition. Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 4 / 16

Matroids Definition For I ⊆ 2 S , M = (S , I ) is a matroid if 1 I � = ∅ , 2 If X ⊆ Y ∈ I then X ∈ I , 3 If X , Y ∈ I with | X | < | Y | then ∃ y ∈ Y \ X such that X ∪ y ∈ I . Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 5 / 16

Matroids Definition For I ⊆ 2 S , M = (S , I ) is a matroid if 1 I � = ∅ , 2 If X ⊆ Y ∈ I then X ∈ I , 3 If X , Y ∈ I with | X | < | Y | then ∃ y ∈ Y \ X such that X ∪ y ∈ I . Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 5 / 16

Matroids Definition For I ⊆ 2 S , M = (S , I ) is a matroid if 1 I � = ∅ , 2 If X ⊆ Y ∈ I then X ∈ I , 3 If X , Y ∈ I with | X | < | Y | then ∃ y ∈ Y \ X such that X ∪ y ∈ I . Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 5 / 16

Matroids Definition For I ⊆ 2 S , M = (S , I ) is a matroid if 1 I � = ∅ , 2 If X ⊆ Y ∈ I then X ∈ I , 3 If X , Y ∈ I with | X | < | Y | then ∃ y ∈ Y \ X such that X ∪ y ∈ I . Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 5 / 16

Matroids Definition For I ⊆ 2 S , M = (S , I ) is a matroid if 1 I � = ∅ , 2 If X ⊆ Y ∈ I then X ∈ I , 3 If X , Y ∈ I with | X | < | Y | then ∃ y ∈ Y \ X such that X ∪ y ∈ I . Examples 1 Sets of linearly independent vectors in a vector space, 2 Edge-sets of forests of a graph, 3 U n , k = { X ⊆ S : | X | ≤ k } where | S | = n , Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 5 / 16

Matroids Definition For I ⊆ 2 S , M = (S , I ) is a matroid if 1 I � = ∅ , 2 If X ⊆ Y ∈ I then X ∈ I , 3 If X , Y ∈ I with | X | < | Y | then ∃ y ∈ Y \ X such that X ∪ y ∈ I . Examples 1 Sets of linearly independent vectors in a vector space, 2 Edge-sets of forests of a graph, 3 U n , k = { X ⊆ S : | X | ≤ k } where | S | = n , Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 5 / 16

Matroids Definition For I ⊆ 2 S , M = (S , I ) is a matroid if 1 I � = ∅ , 2 If X ⊆ Y ∈ I then X ∈ I , 3 If X , Y ∈ I with | X | < | Y | then ∃ y ∈ Y \ X such that X ∪ y ∈ I . Examples 1 Sets of linearly independent vectors in a vector space, 2 Edge-sets of forests of a graph, 3 U n , k = { X ⊆ S : | X | ≤ k } where | S | = n , Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 5 / 16

Matroids Definition For I ⊆ 2 S , M = (S , I ) is a matroid if 1 I � = ∅ , 2 If X ⊆ Y ∈ I then X ∈ I , 3 If X , Y ∈ I with | X | < | Y | then ∃ y ∈ Y \ X such that X ∪ y ∈ I . Examples 1 Sets of linearly independent vectors in a vector space, 2 Edge-sets of forests of a graph, 3 U n , k = { X ⊆ S : | X | ≤ k } where | S | = n , free matroid = U n , n . Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 5 / 16

Matroids Notion 1 independent sets = I , any subset of an independent set is independent, 1 2 base = maximal independent set, all basis are of the same size, 1 3 rank function : r ( X ) = max {| Y | : Y ∈ I , Y ⊆ X } . non-decreasing, 1 submodular, 2 X ∈ I if and only if r ( X ) = | X | . 3 Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 6 / 16

Matroids Notion 1 independent sets = I , any subset of an independent set is independent, 1 2 base = maximal independent set, all basis are of the same size, 1 3 rank function : r ( X ) = max {| Y | : Y ∈ I , Y ⊆ X } . non-decreasing, 1 submodular, 2 X ∈ I if and only if r ( X ) = | X | . 3 Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 6 / 16

Matroids Notion 1 independent sets = I , any subset of an independent set is independent, 1 2 base = maximal independent set, all basis are of the same size, 1 3 rank function : r ( X ) = max {| Y | : Y ∈ I , Y ⊆ X } . non-decreasing, 1 submodular, 2 X ∈ I if and only if r ( X ) = | X | . 3 Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 6 / 16

Matroids Notion 1 independent sets = I , any subset of an independent set is independent, 1 2 base = maximal independent set, all basis are of the same size, 1 3 rank function : r ( X ) = max {| Y | : Y ∈ I , Y ⊆ X } . non-decreasing, 1 submodular, 2 X ∈ I if and only if r ( X ) = | X | . 3 Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 6 / 16

Matroids Notion 1 independent sets = I , any subset of an independent set is independent, 1 2 base = maximal independent set, all basis are of the same size, 1 3 rank function : r ( X ) = max {| Y | : Y ∈ I , Y ⊆ X } . non-decreasing, 1 submodular, 2 X ∈ I if and only if r ( X ) = | X | . 3 Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 6 / 16

Matroids Notion 1 independent sets = I , any subset of an independent set is independent, 1 2 base = maximal independent set, all basis are of the same size, 1 3 rank function : r ( X ) = max {| Y | : Y ∈ I , Y ⊆ X } . non-decreasing, 1 submodular, 2 X ∈ I if and only if r ( X ) = | X | . 3 Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 6 / 16

Matroids Notion 1 independent sets = I , any subset of an independent set is independent, 1 2 base = maximal independent set, all basis are of the same size, 1 3 rank function : r ( X ) = max {| Y | : Y ∈ I , Y ⊆ X } . non-decreasing, 1 submodular, 2 X ∈ I if and only if r ( X ) = | X | . 3 Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 6 / 16

Matroids Notion 1 independent sets = I , any subset of an independent set is independent, 1 2 base = maximal independent set, all basis are of the same size, 1 3 rank function : r ( X ) = max {| Y | : Y ∈ I , Y ⊆ X } . non-decreasing, 1 submodular, 2 X ∈ I if and only if r ( X ) = | X | . 3 Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 6 / 16

Matroid-based rooted-graphs Definition A matroid-based rooted-graph is a quadruple ( G , M , S , π ) : 1 G = ( V , E ) undirected graph, 2 M a matroid on a set S = { s 1 , . . . , s t } . 3 π a placement of the elements of S at vertices of V . π ( s 1 ) π ( s 2 ) G S = { s 1 , s 2 , s 3 } M = U 3 , 2 π ( s 3 ) Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 7 / 16

Matroid-based rooted-graphs Definition A matroid-based rooted-graph is a quadruple ( G , M , S , π ) : 1 G = ( V , E ) undirected graph, 2 M a matroid on a set S = { s 1 , . . . , s t } . 3 π a placement of the elements of S at vertices of V . π ( s 1 ) X π ( s 2 ) S X = { s 1 , s 2 } π ( s 3 ) Notation S X = the elements of S placed at X (= π − 1 ( X )) . Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 7 / 16

M -based packing of rooted-trees Definition A rooted-tree is a pair ( T , s) where 1 T is a tree, 2 s ∈ S, placed at a vertex of T . π ( s 1 ) π ( s 2 ) T 2 T 1 T 3 π ( s 3 ) Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 8 / 16

M -based packing of rooted-trees Definition A rooted-tree is a pair ( T , s) where 1 T is a tree, 2 s ∈ S, placed at a vertex of T . π ( s 1 ) π ( s 2 ) T 2 T 1 T 3 π ( s 3 ) Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 8 / 16

M -based packing of rooted-trees Definition A rooted-tree is a pair ( T , s) where 1 T is a tree, 2 s ∈ S, placed at a vertex of T . π ( s 1 ) π ( s 2 ) T 2 T 1 T 3 π ( s 3 ) Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 8 / 16

M -based packing of rooted-trees Definition A rooted-tree is a pair ( T , s) where 1 T is a tree, 2 s ∈ S, placed at a vertex of T . π ( s 1 ) π ( s 2 ) T 2 T 1 T 3 π ( s 3 ) Definition A packing { ( T 1 , s 1 ) , . . . , ( T | S | , s | S | ) } of rooted-trees is called M -based if Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 8 / 16

M -based packing of rooted-trees Definition A rooted-tree is a pair ( T , s) where 1 T is a tree, 2 s ∈ S, placed at a vertex of T . π ( s 1 ) π ( s 2 ) T 2 T 1 T 3 π ( s 3 ) Definition A packing { ( T 1 , s 1 ) , . . . , ( T | S | , s | S | ) } of rooted-trees is called M -based if { s i ∈ S : v ∈ V ( T i ) } forms a base of M for every v ∈ V . Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 8 / 16

M -based packing of rooted-trees Definition A packing { ( T 1 , s 1 ) , . . . , ( T | S | , s | S | ) } of rooted-trees is called M -based if { s i ∈ S : v ∈ V ( T i ) } forms a base of M for every v ∈ V . Remark For the free matroid M with all k roots at a vertex r , matroid-based packing of rooted-trees ⇐ ⇒ packing of k spanning trees. Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 8 / 16

M -based packing of rooted-trees Definition A packing { ( T 1 , s 1 ) , . . . , ( T | S | , s | S | ) } of rooted-trees is called M -based if { s i ∈ S : v ∈ V ( T i ) } forms a base of M for every v ∈ V . Remark For the free matroid M with all k roots at a vertex r , matroid-based packing of rooted-trees ⇐ ⇒ packing of k spanning trees. Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 8 / 16

M -based packing of rooted-trees Definition A packing { ( T 1 , s 1 ) , . . . , ( T | S | , s | S | ) } of rooted-trees is called M -based if { s i ∈ S : v ∈ V ( T i ) } forms a base of M for every v ∈ V . Remark For the free matroid M with all k roots at a vertex r , matroid-based packing of rooted-trees ⇐ ⇒ packing of k spanning trees. Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 8 / 16

M -based packing of rooted-trees Definition A packing { ( T 1 , s 1 ) , . . . , ( T | S | , s | S | ) } of rooted-trees is called M -based if { s i ∈ S : v ∈ V ( T i ) } forms a base of M for every v ∈ V . Definitions 1 π is M -independent if for every v ∈ V , S v is independent in M . 2 ( G , M , S , π ) is partition-connected if for every partition P of V , e G ( P ) ≥ � X ∈P ( r M (S) − r M (S X )) . Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 8 / 16

M -based packing of rooted-trees Definition A packing { ( T 1 , s 1 ) , . . . , ( T | S | , s | S | ) } of rooted-trees is called M -based if { s i ∈ S : v ∈ V ( T i ) } forms a base of M for every v ∈ V . Definitions 1 π is M -independent if for every v ∈ V , S v is independent in M . 2 ( G , M , S , π ) is partition-connected if for every partition P of V , e G ( P ) ≥ � X ∈P ( r M (S) − r M (S X )) . Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 8 / 16

M -based packing of rooted-trees Definition A packing { ( T 1 , s 1 ) , . . . , ( T | S | , s | S | ) } of rooted-trees is called M -based if { s i ∈ S : v ∈ V ( T i ) } forms a base of M for every v ∈ V . Definitions 1 π is M -independent if for every v ∈ V , S v is independent in M . 2 ( G , M , S , π ) is partition-connected if for every partition P of V , e G ( P ) ≥ � X ∈P ( r M (S) − r M (S X )) . Theorem (Katoh, Tanigawa 2012) Let ( G , M , S , π ) be a matroid-based rooted-graph. There is a matroid-based packing of rooted-trees in ( G , M , S , π ) ⇐ ⇒ π is M -independent and ( G , M , S , π ) is partition-connected. Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 8 / 16

M -based packing of rooted-trees Definition A packing { ( T 1 , s 1 ) , . . . , ( T | S | , s | S | ) } of rooted-trees is called M -based if { s i ∈ S : v ∈ V ( T i ) } forms a base of M for every v ∈ V . Definitions 1 π is M -independent if for every v ∈ V , S v is independent in M . 2 ( G , M , S , π ) is partition-connected if for every partition P of V , e G ( P ) ≥ � X ∈P ( r M (S) − r M (S X )) . Theorem (Katoh, Tanigawa 2012) Let ( G , M , S , π ) be a matroid-based rooted-graph. There is a matroid-based packing of rooted-trees in ( G , M , S , π ) ⇐ ⇒ π is M -independent and ( G , M , S , π ) is partition-connected. Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 8 / 16

M -based packing of rooted-trees Definition A packing { ( T 1 , s 1 ) , . . . , ( T | S | , s | S | ) } of rooted-trees is called M -based if { s i ∈ S : v ∈ V ( T i ) } forms a base of M for every v ∈ V . Definitions 1 π is M -independent if for every v ∈ V , S v is independent in M . 2 ( G , M , S , π ) is partition-connected if for every partition P of V , e G ( P ) ≥ � X ∈P ( r M (S) − r M (S X )) . Theorem (Katoh, Tanigawa 2012) Let ( G , M , S , π ) be a matroid-based rooted-graph. There is a matroid-based packing of rooted-trees in ( G , M , S , π ) ⇐ ⇒ π is M -independent and ( G , M , S , π ) is partition-connected. Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 8 / 16

M -based packing of rooted-arborescences Definition A rooted-arborescence is a pair ( T , s) where 1 T is an r -arborescence, 2 s ∈ S, placed at r . π ( s 1 ) π ( s 2 ) T 2 T 1 T 3 π ( s 3 ) Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 9 / 16

M -based packing of rooted-arborescences Definition A rooted-arborescence is a pair ( T , s) where 1 T is an r -arborescence, 2 s ∈ S, placed at r . π ( s 1 ) π ( s 2 ) T 2 T 1 T 3 π ( s 3 ) Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 9 / 16

M -based packing of rooted-arborescences Definition A rooted-arborescence is a pair ( T , s) where 1 T is an r -arborescence, 2 s ∈ S, placed at r . π ( s 1 ) π ( s 2 ) T 2 T 1 T 3 π ( s 3 ) Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 9 / 16

M -based packing of rooted-arborescences Definition A rooted-arborescence is a pair ( T , s) where 1 T is an r -arborescence, 2 s ∈ S, placed at r . π ( s 1 ) π ( s 2 ) T 2 T 1 T 3 π ( s 3 ) Definition A packing { ( T 1 , s 1 ) , . . . , ( T | S | , s | S | ) } of rooted-arborescences is M -based if Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 9 / 16

M -based packing of rooted-arborescences Definition A rooted-arborescence is a pair ( T , s) where 1 T is an r -arborescence, 2 s ∈ S, placed at r . π ( s 1 ) π ( s 2 ) T 2 T 1 T 3 π ( s 3 ) Definition A packing { ( T 1 , s 1 ) , . . . , ( T | S | , s | S | ) } of rooted-arborescences is M -based if { s i ∈ S : v ∈ V ( T i ) } forms a base of M for every v ∈ V . Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 9 / 16

M -based packing of rooted-arborescences Definition A packing { ( T 1 , s 1 ) , . . . , ( T | S | , s | S | ) } of rooted-arborescences is M -based if { s i ∈ S : v ∈ V ( T i ) } forms a base of M for every v ∈ V . Remark For the free matroid M with all k roots at a vertex r , matroid-based packing of rooted-arborescences ⇐ ⇒ packing of k spanning r -arborescences. Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 9 / 16

M -based packing of rooted-arborescences Definition A packing { ( T 1 , s 1 ) , . . . , ( T | S | , s | S | ) } of rooted-arborescences is M -based if { s i ∈ S : v ∈ V ( T i ) } forms a base of M for every v ∈ V . Remark For the free matroid M with all k roots at a vertex r , matroid-based packing of rooted-arborescences ⇐ ⇒ packing of k spanning r -arborescences. Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 9 / 16

M -based packing of rooted-arborescences Definition A packing { ( T 1 , s 1 ) , . . . , ( T | S | , s | S | ) } of rooted-arborescences is M -based if { s i ∈ S : v ∈ V ( T i ) } forms a base of M for every v ∈ V . Remark For the free matroid M with all k roots at a vertex r , matroid-based packing of rooted-arborescences ⇐ ⇒ packing of k spanning r -arborescences. Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 9 / 16

M -based packing of rooted-arborescences Definition A packing { ( T 1 , s 1 ) , . . . , ( T | S | , s | S | ) } of rooted-arborescences is M -based if { s i ∈ S : v ∈ V ( T i ) } forms a base of M for every v ∈ V . Definition ( D , M , S , π ) is rooted-connected if for every ∅ � = X ⊆ V , ρ D ( X ) ≥ r M (S) − r M (S X ) . Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 9 / 16

M -based packing of rooted-arborescences Definition A packing { ( T 1 , s 1 ) , . . . , ( T | S | , s | S | ) } of rooted-arborescences is M -based if { s i ∈ S : v ∈ V ( T i ) } forms a base of M for every v ∈ V . Definition ( D , M , S , π ) is rooted-connected if for every ∅ � = X ⊆ V , ρ D ( X ) ≥ r M (S) − r M (S X ) . Theorem (Durand de Gevigney, Nguyen, Szigeti 2012) Let ( D , M , S , π ) be a matroid-based rooted-digraph. There is a matroid-based packing of rooted-arborescences in ( D , M , S , π ) ⇐ ⇒ π is M -independent and ( D , M , S , π ) is rooted-connected. Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 9 / 16

M -based packing of rooted-arborescences Definition A packing { ( T 1 , s 1 ) , . . . , ( T | S | , s | S | ) } of rooted-arborescences is M -based if { s i ∈ S : v ∈ V ( T i ) } forms a base of M for every v ∈ V . Definition ( D , M , S , π ) is rooted-connected if for every ∅ � = X ⊆ V , ρ D ( X ) ≥ r M (S) − r M (S X ) . Theorem (Durand de Gevigney, Nguyen, Szigeti 2012) Let ( D , M , S , π ) be a matroid-based rooted-digraph. There is a matroid-based packing of rooted-arborescences in ( D , M , S , π ) ⇐ ⇒ π is M -independent and ( D , M , S , π ) is rooted-connected. Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 9 / 16

M -based packing of rooted-arborescences Definition A packing { ( T 1 , s 1 ) , . . . , ( T | S | , s | S | ) } of rooted-arborescences is M -based if { s i ∈ S : v ∈ V ( T i ) } forms a base of M for every v ∈ V . Definition ( D , M , S , π ) is rooted-connected if for every ∅ � = X ⊆ V , ρ D ( X ) ≥ r M (S) − r M (S X ) . Theorem (Durand de Gevigney, Nguyen, Szigeti 2012) Let ( D , M , S , π ) be a matroid-based rooted-digraph. There is a matroid-based packing of rooted-arborescences in ( D , M , S , π ) ⇐ ⇒ π is M -independent and ( D , M , S , π ) is rooted-connected. Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 9 / 16

Proof of necessity Let { ( T 1 , s 1 ) , . . . , ( T | S | , s | S | ) } be a matroid-based packing of rooted-arborescences in ( D , M , S , π ) and v ∈ X ⊆ V . Let B = { s i ∈ S : v ∈ V ( T i ) } , B 1 = B ∩ S X and B 2 = B \ B 1 . Since S v ⊆ B 1 ⊆ B is a base of M , π is M -independent. Since, for each root s i in B 2 , there exists an arc of T i that enters X and the arborescences are arc-disjoint, ρ D ( X ) ≥ | B 2 | = | B | − | B 1 | = r M (S) − r M ( B 1 ) ≥ r M (S) − r M (S X ) that is ( D , M , S , π ) is rooted-connected. Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 10 / 16

Proof of necessity Let { ( T 1 , s 1 ) , . . . , ( T | S | , s | S | ) } be a matroid-based packing of rooted-arborescences in ( D , M , S , π ) and v ∈ X ⊆ V . Let B = { s i ∈ S : v ∈ V ( T i ) } , B 1 = B ∩ S X and B 2 = B \ B 1 . Since S v ⊆ B 1 ⊆ B is a base of M , π is M -independent. Since, for each root s i in B 2 , there exists an arc of T i that enters X and the arborescences are arc-disjoint, ρ D ( X ) ≥ | B 2 | = | B | − | B 1 | = r M (S) − r M ( B 1 ) ≥ r M (S) − r M (S X ) that is ( D , M , S , π ) is rooted-connected. π ( s | B 2 | ) π ( s 1 ) π ( s ℓ ) π ( s | B 1 | ) π ( s j ) X π ( s 2 ) Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 10 / 16

Proof of necessity Let { ( T 1 , s 1 ) , . . . , ( T | S | , s | S | ) } be a matroid-based packing of rooted-arborescences in ( D , M , S , π ) and v ∈ X ⊆ V . Let B = { s i ∈ S : v ∈ V ( T i ) } , B 1 = B ∩ S X and B 2 = B \ B 1 . Since S v ⊆ B 1 ⊆ B is a base of M , π is M -independent. Since, for each root s i in B 2 , there exists an arc of T i that enters X and the arborescences are arc-disjoint, ρ D ( X ) ≥ | B 2 | = | B | − | B 1 | = r M (S) − r M ( B 1 ) ≥ r M (S) − r M (S X ) that is ( D , M , S , π ) is rooted-connected. π ( s | B 2 | ) π ( s 1 ) π ( s ℓ ) π ( s | B 1 | ) π ( s j ) X π ( s 2 ) Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 10 / 16

Proof of necessity Let { ( T 1 , s 1 ) , . . . , ( T | S | , s | S | ) } be a matroid-based packing of rooted-arborescences in ( D , M , S , π ) and v ∈ X ⊆ V . Let B = { s i ∈ S : v ∈ V ( T i ) } , B 1 = B ∩ S X and B 2 = B \ B 1 . Since S v ⊆ B 1 ⊆ B is a base of M , π is M -independent. Since, for each root s i in B 2 , there exists an arc of T i that enters X and the arborescences are arc-disjoint, ρ D ( X ) ≥ | B 2 | = | B | − | B 1 | = r M (S) − r M ( B 1 ) ≥ r M (S) − r M (S X ) that is ( D , M , S , π ) is rooted-connected. π ( s | B 2 | ) π ( s 1 ) π ( s ℓ ) π ( s | B 1 | ) π ( s j ) X π ( s 2 ) Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 10 / 16

Orientation results Theorem (Frank 1980) Let G = ( V , E ) be an undirected graph and h : 2 V → Z + an intersecting supermodular non-increasing set-function. There is an orientation D of G s. t. ρ D ( X ) ≥ h ( X ) ∀ ∅ � = X ⊂ V ⇐ ⇒ e G ( P ) ≥ � X ∈P h ( X ) for every partition P of V . Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 11 / 16

Orientation results Theorem (Frank 1980) Let G = ( V , E ) be an undirected graph and h : 2 V → Z + an intersecting supermodular non-increasing set-function. There is an orientation D of G s. t. ρ D ( X ) ≥ h ( X ) ∀ ∅ � = X ⊂ V ⇐ ⇒ e G ( P ) ≥ � X ∈P h ( X ) for every partition P of V . Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 11 / 16

Orientation results Theorem (Frank 1980) Let G = ( V , E ) be an undirected graph and h : 2 V → Z + an intersecting supermodular non-increasing set-function. There is an orientation D of G s. t. ρ D ( X ) ≥ h ( X ) ∀ ∅ � = X ⊂ V ⇐ ⇒ e G ( P ) ≥ � X ∈P h ( X ) for every partition P of V . Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 11 / 16

Orientation results Theorem (Frank 1980) Let G = ( V , E ) be an undirected graph and h : 2 V → Z + an intersecting supermodular non-increasing set-function. There is an orientation D of G s. t. ρ D ( X ) ≥ h ( X ) ∀ ∅ � = X ⊂ V ⇐ ⇒ e G ( P ) ≥ � X ∈P h ( X ) for every partition P of V . Applying for h ( X ) = r M (S) − r M (S X ) provides Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 11 / 16

Orientation results Theorem (Frank 1980) Let G = ( V , E ) be an undirected graph and h : 2 V → Z + an intersecting supermodular non-increasing set-function. There is an orientation D of G s. t. ρ D ( X ) ≥ h ( X ) ∀ ∅ � = X ⊂ V ⇐ ⇒ e G ( P ) ≥ � X ∈P h ( X ) for every partition P of V . Applying for h ( X ) = r M (S) − r M (S X ) provides Corollary Let ( G , M , S , π ) be a matroid-based rooted-graph. There is an orientation D of G s. t. ( D , M , S , π ) is rooted-connected ⇐ ⇒ ( G , M , S , π ) is partition-connected. Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 11 / 16

Orientation results Theorem (Frank 1980) Let G = ( V , E ) be an undirected graph and h : 2 V → Z + an intersecting supermodular non-increasing set-function. There is an orientation D of G s. t. ρ D ( X ) ≥ h ( X ) ∀ ∅ � = X ⊂ V ⇐ ⇒ e G ( P ) ≥ � X ∈P h ( X ) for every partition P of V . Applying for h ( X ) = r M (S) − r M (S X ) provides Corollary Let ( G , M , S , π ) be a matroid-based rooted-graph. There is an orientation D of G s. t. ( D , M , S , π ) is rooted-connected ⇐ ⇒ ( G , M , S , π ) is partition-connected. Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 11 / 16

Orientation results Theorem (Frank 1980) Let G = ( V , E ) be an undirected graph and h : 2 V → Z + an intersecting supermodular non-increasing set-function. There is an orientation D of G s. t. ρ D ( X ) ≥ h ( X ) ∀ ∅ � = X ⊂ V ⇐ ⇒ e G ( P ) ≥ � X ∈P h ( X ) for every partition P of V . Applying for h ( X ) = r M (S) − r M (S X ) provides Corollary Let ( G , M , S , π ) be a matroid-based rooted-graph. There is an orientation D of G s. t. ( D , M , S , π ) is rooted-connected ⇐ ⇒ ( G , M , S , π ) is partition-connected. Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 11 / 16

Plan executed Theorem (Katoh, Tanigawa 2012) ∃ a matroid-based packing of rooted-trees in ( G , M , S , π ) ⇐ ⇒ π is M -independent and ( G , M , S , π ) is partition-connected. Theorem (Durand de Gevigney, Nguyen, Szigeti 2012) ∃ a matroid-based packing of rooted-arborescences in ( D , M , S , π ) ⇐ ⇒ π is M -independent and ( D , M , S , π ) is rooted-connected. Theorem (Frank 1980) ∃ an orientation D of G s. t. ( D , M , S , π ) is rooted-connected ⇐ ⇒ ( G , M , S , π ) is partition-connected. Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 12 / 16

Plan executed Theorem (Katoh, Tanigawa 2012) ∃ a matroid-based packing of rooted-trees in ( G , M , S , π ) ⇐ ⇒ π is M -independent and ( G , M , S , π ) is partition-connected. Theorem (Durand de Gevigney, Nguyen, Szigeti 2012) ∃ a matroid-based packing of rooted-arborescences in ( D , M , S , π ) ⇐ ⇒ π is M -independent and ( D , M , S , π ) is rooted-connected. Theorem (Frank 1980) ∃ an orientation D of G s. t. ( D , M , S , π ) is rooted-connected ⇐ ⇒ ( G , M , S , π ) is partition-connected. Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 12 / 16

Plan executed Theorem (Katoh, Tanigawa 2012) ∃ a matroid-based packing of rooted-trees in ( G , M , S , π ) ⇐ ⇒ π is M -independent and ( G , M , S , π ) is partition-connected. Theorem (Durand de Gevigney, Nguyen, Szigeti 2012) ∃ a matroid-based packing of rooted-arborescences in ( D , M , S , π ) ⇐ ⇒ π is M -independent and ( D , M , S , π ) is rooted-connected. Theorem (Frank 1980) ∃ an orientation D of G s. t. ( D , M , S , π ) is rooted-connected ⇐ ⇒ ( G , M , S , π ) is partition-connected. Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 12 / 16

Plan executed Theorem (Katoh, Tanigawa 2012) ∃ a matroid-based packing of rooted-trees in ( G , M , S , π ) ⇐ ⇒ π is M -independent and ( G , M , S , π ) is partition-connected. Theorem (Durand de Gevigney, Nguyen, Szigeti 2012) ∃ a matroid-based packing of rooted-arborescences in ( D , M , S , π ) ⇐ ⇒ π is M -independent and ( D , M , S , π ) is rooted-connected. Theorem (Frank 1980) ∃ an orientation D of G s. t. ( D , M , S , π ) is rooted-connected ⇐ ⇒ ( G , M , S , π ) is partition-connected. Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 12 / 16

Plan executed Theorem (Katoh, Tanigawa 2012) ∃ a matroid-based packing of rooted-trees in ( G , M , S , π ) ⇐ ⇒ π is M -independent and ( G , M , S , π ) is partition-connected. Theorem (Durand de Gevigney, Nguyen, Szigeti 2012) ∃ a matroid-based packing of rooted-arborescences in ( D , M , S , π ) ⇐ ⇒ π is M -independent and ( D , M , S , π ) is rooted-connected. Theorem (Frank 1980) ∃ an orientation D of G s. t. ( D , M , S , π ) is rooted-connected ⇐ ⇒ ( G , M , S , π ) is partition-connected. Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 12 / 16

Plan executed Theorem (Katoh, Tanigawa 2012) ∃ a matroid-based packing of rooted-trees in ( G , M , S , π ) ⇐ ⇒ π is M -independent and ( G , M , S , π ) is partition-connected. Theorem (Durand de Gevigney, Nguyen, Szigeti 2012) ∃ a matroid-based packing of rooted-arborescences in ( D , M , S , π ) ⇐ ⇒ π is M -independent and ( D , M , S , π ) is rooted-connected. Theorem (Frank 1980) ∃ an orientation D of G s. t. ( D , M , S , π ) is rooted-connected ⇐ ⇒ ( G , M , S , π ) is partition-connected. Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 12 / 16

Plan executed Theorem (Katoh, Tanigawa 2012) ∃ a matroid-based packing of rooted-trees in ( G , M , S , π ) ⇐ ⇒ π is M -independent and ( G , M , S , π ) is partition-connected. Theorem (Durand de Gevigney, Nguyen, Szigeti 2012) ∃ a matroid-based packing of rooted-arborescences in ( D , M , S , π ) ⇐ ⇒ π is M -independent and ( D , M , S , π ) is rooted-connected. Theorem (Frank 1980) ∃ an orientation D of G s. t. ( D , M , S , π ) is rooted-connected ⇐ ⇒ ( G , M , S , π ) is partition-connected. Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 12 / 16

Plan executed Theorem (Katoh, Tanigawa 2012) ∃ a matroid-based packing of rooted-trees in ( G , M , S , π ) ⇐ ⇒ π is M -independent and ( G , M , S , π ) is partition-connected. Theorem (Durand de Gevigney, Nguyen, Szigeti 2012) ∃ a matroid-based packing of rooted-arborescences in ( D , M , S , π ) ⇐ ⇒ π is M -independent and ( D , M , S , π ) is rooted-connected. Theorem (Frank 1980) ∃ an orientation D of G s. t. ( D , M , S , π ) is rooted-connected ⇐ ⇒ ( G , M , S , π ) is partition-connected. Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 12 / 16

Plan executed Theorem (Katoh, Tanigawa 2012) ∃ a matroid-based packing of rooted-trees in ( G , M , S , π ) ⇐ ⇒ π is M -independent and ( G , M , S , π ) is partition-connected. Theorem (Durand de Gevigney, Nguyen, Szigeti 2012) ∃ a matroid-based packing of rooted-arborescences in ( D , M , S , π ) ⇐ ⇒ π is M -independent and ( D , M , S , π ) is rooted-connected. Theorem (Frank 1980) ∃ an orientation D of G s. t. ( D , M , S , π ) is rooted-connected ⇐ ⇒ ( G , M , S , π ) is partition-connected. Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 12 / 16

Plan executed Theorem (Katoh, Tanigawa 2012) ∃ a matroid-based packing of rooted-trees in ( G , M , S , π ) ⇐ ⇒ π is M -independent and ( G , M , S , π ) is partition-connected. Theorem (Durand de Gevigney, Nguyen, Szigeti 2012) ∃ a matroid-based packing of rooted-arborescences in ( D , M , S , π ) ⇐ ⇒ π is M -independent and ( D , M , S , π ) is rooted-connected. Theorem (Frank 1980) ∃ an orientation D of G s. t. ( D , M , S , π ) is rooted-connected ⇐ ⇒ ( G , M , S , π ) is partition-connected. Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 12 / 16

About the proofs Theorem (Katoh, Tanigawa 2012) ∃ a matroid-based packing of rooted-trees in ( G , M , S , π ) ⇐ ⇒ π is M -independent and ( G , M , S , π ) is partition-connected. Theorem (Durand de Gevigney, Nguyen, Szigeti 2012) ∃ a matroid-based packing of rooted-arborescences in ( D , M , S , π ) ⇐ ⇒ π is M -independent and ( D , M , S , π ) is rooted-connected. Theorem (Frank 1980) ∃ an orientation D of G s. t. ( D , M , S , π ) is rooted-connected ⇐ ⇒ ( G , M , S , π ) is partition-connected. Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 13 / 16

About the proofs Theorem (Katoh, Tanigawa 2012) ∃ a matroid-based packing of rooted-trees in ( G , M , S , π ) ⇐ ⇒ 8 pages π is M -independent and ( G , M , S , π ) is partition-connected. Theorem (Durand de Gevigney, Nguyen, Szigeti 2012) ∃ a matroid-based packing of rooted-arborescences in ( D , M , S , π ) ⇐ ⇒ π is M -independent and ( D , M , S , π ) is rooted-connected. Theorem (Frank 1980) ∃ an orientation D of G s. t. ( D , M , S , π ) is rooted-connected ⇐ ⇒ ( G , M , S , π ) is partition-connected. Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 13 / 16

About the proofs Theorem (Katoh, Tanigawa 2012) ∃ a matroid-based packing of rooted-trees in ( G , M , S , π ) ⇐ ⇒ 8 pages π is M -independent and ( G , M , S , π ) is partition-connected. Theorem (Durand de Gevigney, Nguyen, Szigeti 2012) ∃ a matroid-based packing of rooted-arborescences in ( D , M , S , π ) ⇐ ⇒ 2 pages π is M -independent and ( D , M , S , π ) is rooted-connected. Theorem (Frank 1980) ∃ an orientation D of G s. t. ( D , M , S , π ) is rooted-connected ⇐ ⇒ ( G , M , S , π ) is partition-connected. Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 13 / 16

About the proofs Theorem (Katoh, Tanigawa 2012) ∃ a matroid-based packing of rooted-trees in ( G , M , S , π ) ⇐ ⇒ 8 pages π is M -independent and ( G , M , S , π ) is partition-connected. Theorem (Durand de Gevigney, Nguyen, Szigeti 2012) ∃ a matroid-based packing of rooted-arborescences in ( D , M , S , π ) ⇐ ⇒ 2 pages π is M -independent and ( D , M , S , π ) is rooted-connected. Theorem (Frank 1980) ∃ an orientation D of G s. t. ( D , M , S , π ) is rooted-connected ⇐ ⇒ 4 pages ( G , M , S , π ) is partition-connected. Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences January 2013 13 / 16