Binary matroids and Fundamental graphs O-joung Kwon May 3, 2012 Binary matroids and Fundamental graphs O-joung Kwon May 3, 2012 Matroid E : set, I : a set of subsets of E. A pair M = (E; I) is a matroid on E if it satisfies the following three axioms. (T1) I= 6 ;. (T2) If X 2 I and Y ⊆ X, then Y 2 I. (T3) Let X; Y 2 I and jXj = jY j + 1. Then there exists x 2 X n Y such that Y [ fxg 2 I. X 2 I : an independent set of M. An independent set X of M is a base of M if it is a maximal independent set. Matroid E : set, I : a set of subsets of E. A pair M = (E; I) is a matroid on E if it satisfies the following three axioms. (T1) I= 6 ;. (T2) If X 2 I and Y ⊆ X, then Y 2 I. (T3) Let X; Y 2 I and jXj = jY j + 1. Then there exists x 2 X n Y such that Y [ fxg 2 I. X 2 I : an independent set of M. An independent set X of M is a base of M if it is a maximal independent set. Matroid E : set, I : a set of subsets of E. A pair M = (E; I) is a matroid on E if it satisfies the following three axioms. (T1) I= 6 ;. (T2) If X 2 I and Y ⊆ X, then Y 2 I. (T3) Let X; Y 2 I and jXj = jY j + 1. Then there exists x 2 X n Y such that Y [ fxg 2 I. X 2 I : an independent set of M. An independent set X of M is a base of M if it is a maximal independent set. Matroid E : set, I : a set of subsets of E. A pair M = (E; I) is a matroid on E if it satisfies the following three axioms. (T1) I= 6 ;. (T2) If X 2 I and Y ⊆ X, then Y 2 I. (T3) Let X; Y 2 I and jXj = jY j + 1. Then there exists x 2 X n Y such that Y [ fxg 2 I. X 2 I : an independent set of M. An independent set X of M is a base of M if it is a maximal independent set. Matroid E : set, I : a set of subsets of E. A pair M = (E; I) is a matroid on E if it satisfies the following three axioms. (T1) I= 6 ;. (T2) If X 2 I and Y ⊆ X, then Y 2 I. (T3) Let X; Y 2 I and jXj = jY j + 1. Then there exists x 2 X n Y such that Y [ fxg 2 I. X 2 I : an independent set of M. An independent set X of M is a base of M if it is a maximal independent set. Binary Matroid Let A be a V × W (0; 1)-matrix. The binary matroid M(A) of A is defined as (W; fU ⊆ W : U represent a linearly independent set of Ag). By the definition, a base of M(A) forms a maximally lin. independent set of column space of A. a b c d e 1 0 0 1 1! A = 0 1 0 1 1 0 0 1 0 1 fa; b; cg 2 I : a base. Graphic Matroid Let G be a graph. The graphic matroid M(G) of G is defined as M(G) = (E(G); fU ⊆ E(G): U forms a forest in Gg). By the definition, a base of M(G) forms a spanning tree of G. a e f G b d c fa; f; e; dg 2 I : a base Graphic Matroid 1 a e 2 f 5 G b d 3 c 4 a b c d e f 10 1 0 0 0 1 11 2B 1 1 0 0 0 0C I = B C G 3B 0 1 1 0 0 1C 4@ 0 0 1 1 0 0A 5 0 0 0 1 1 0 Graphic and Binary Matroid Theorem Let G be a graph and IG be the vertex-edge(V × E) incidence matrix of G. Then M(IG) = M(G). Therefore every graphic matroid is binary. 0. Problem. f f g G=e G=e n g Graph minor Let G = (V; E) be a graph. • G=e Contraction an edge e 2 E • G n e Deletion an edge e 2 E • G n v Deletion a vertex v 2 V f e g G f f g G=e G=e n g Graph minor Let G = (V; E) be a graph. • G=e Contraction an edge e 2 E • G n e Deletion an edge e 2 E • G n v Deletion a vertex v 2 V f e g G f f g G=e G=e n g Graph minor Let G = (V; E) be a graph. • G=e Contraction an edge e 2 E • G n e Deletion an edge e 2 E • G n v Deletion a vertex v 2 V f e g G f G=e n g Graph minor Let G = (V; E) be a graph. • G=e Contraction an edge e 2 E • G n e Deletion an edge e 2 E • G n v Deletion a vertex v 2 V f f e g g G G=e Graph minor Let G = (V; E) be a graph. • G=e Contraction an edge e 2 E • G n e Deletion an edge e 2 E • G n v Deletion a vertex v 2 V f f f e g g G G=e G=e n g No tree-width: 3 tree-width: 2 Graph minor Does the right graph has the left graph as a minor??? tree-width: 3 tree-width: 2 Graph minor Does the right graph has the left graph as a minor??? No Graph minor Does the right graph has the left graph as a minor??? No tree-width: 3 tree-width: 2 Pivot-minors of graphs We are interested in another operation. • G ^ uv Pivot-operation on an edge uv 2 E But in this talk, every graph has no triangle. C does not appear. Pivot-minors of graphs We are interested in another operation. • G ^ uv Pivot-operation on an edge uv 2 E But in this talk, every graph has no triangle. C does not appear. Pivot-minors of graphs We are interested in another operation. • G ^ uv Pivot-operation on an edge uv 2 E But in this talk, every graph has no triangle. C does not appear. Pivot-minors of graphs We are interested in another operation. • G ^ uv Pivot-operation on an edge uv 2 E But in this talk, every graph has no triangle. C does not appear. e e G G ^ e Pivot-minors of graphs For example, • H : a pivot-minor of a graph G if H is obtained from G by applying a sequence of pivoting edges and vertex deletions. e G ^ e Pivot-minors of graphs For example, e G • H : a pivot-minor of a graph G if H is obtained from G by applying a sequence of pivoting edges and vertex deletions. Pivot-minors of graphs For example, e e G G ^ e • H : a pivot-minor of a graph G if H is obtained from G by applying a sequence of pivoting edges and vertex deletions. Pivot-minors of graphs For example, e e G G ^ e • H : a pivot-minor of a graph G if H is obtained from G by applying a sequence of pivoting edges and vertex deletions. Pivot-minors of graphs For example, e e G G ^ e • H : a pivot-minor of a graph G if H is obtained from G by applying a sequence of pivoting edges and vertex deletions. v0 v2 v1 v3 v4 v5 v6 v7 v0 v2 v1 v4 v3 v5 v6 v7 v0 v2 v1 v4 v3 v6 v5 v7 v0 v2 v4 v6 v7 Example ^v1v2 ^ v3v4 ^ v5v6 n fv1; v3; v5g v0 v1 v2 v3 v4 v5 v6 v7 v0 v2 v1 v4 v3 v5 v6 v7 v0 v2 v1 v4 v3 v6 v5 v7 v0 v2 v4 v6 v7 Example ^v1v2 ^ v3v4 ^ v5v6 n fv1; v3; v5g v0 v1 v2 v3 v4 v5 v6 v7 v0 v2 v1 v3 v4 v5 v6 v7 v0 v2 v1 v4 v3 v6 v5 v7 v0 v2 v4 v6 v7 Example ^v1v2 ^ v3v4 ^ v5v6 n fv1; v3; v5g v0 v1 v2 v3 v4 v5 v6 v7 v0 v2 v1 v3 v4 v5 v6 v7 v0 v2 v1 v4 v3 v5 v6 v7 v0 v2 v4 v6 v7 Example ^v1v2 ^ v3v4 ^ v5v6 n fv1; v3; v5g v0 v1 v2 v3 v4 v5 v6 v7 v0 v2 v1 v3 v4 v5 v6 v7 v0 v2 v1 v4 v3 v5 v6 v7 v0 v2 v1 v4 v3 v6 v5 v7 Example ^v1v2 ^ v3v4 ^ v5v6 n fv1; v3; v5g v0 v1 v2 v3 v4 v5 v6 v7 v0 v2 v1 v3 v4 v5 v6 v7 v0 v2 v1 v4 v3 v5 v6 v7 v0 v2 v1 v4 v3 v6 v5 v7 v0 v2 v4 v6 v7 Complete binary tree The incidence graph of a complete binary tree rank-width: 1 rank-width: 1 k linear r-w: 2 linear rank-width: b 2 c + 1 Question Could the incidence graph of a complete binary tree have the B6 graph as a pivot-minor??? a b B6 graph k linear r-w: 2 linear rank-width: b 2 c + 1 Question Could the incidence graph of a complete binary tree have the B6 graph as a pivot-minor??? a b B6 graph rank-width: 1 rank-width: 1 Question Could the incidence graph of a complete binary tree have the B6 graph as a pivot-minor??? a b B6 graph rank-width: 1 rank-width: 1 k linear r-w: 2 linear rank-width: b 2 c + 1 We have two proofs of the theorem.