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 ∈ I and Y ⊆ X, then Y ∈ I. (T3) Let X,Y ∈ I and |X| = |Y | + 1. Then there exists x ∈ X \ Y such that Y ∪ {x} ∈ I. X ∈ 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 ∈ I and Y ⊆ X, then Y ∈ I. (T3) Let X,Y ∈ I and |X| = |Y | + 1. Then there exists x ∈ X \ Y such that Y ∪ {x} ∈ I. X ∈ 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 ∈ I and Y ⊆ X, then Y ∈ I. (T3) Let X,Y ∈ I and |X| = |Y | + 1. Then there exists x ∈ X \ Y such that Y ∪ {x} ∈ I. X ∈ 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 ∈ I and Y ⊆ X, then Y ∈ I. (T3) Let X,Y ∈ I and |X| = |Y | + 1. Then there exists x ∈ X \ Y such that Y ∪ {x} ∈ I. X ∈ 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 ∈ I and Y ⊆ X, then Y ∈ I. (T3) Let X,Y ∈ I and |X| = |Y | + 1. Then there exists x ∈ X \ Y such that Y ∪ {x} ∈ I. X ∈ 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, {U ⊆ W : U represent a linearly independent set of A}).

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

{a, b, c} ∈ I : a base. Graphic Matroid

Let G be a graph. The graphic matroid M(G) of G is defined as M(G) = (E(G), {U ⊆ E(G): U forms a forest in G}).

By the definition, a base of M(G) forms a spanning tree of G.

a e

f G b d c

{a, f, e, d} ∈ I : a base Graphic Matroid

1 a e

2 f 5 G b d 3 c 4

a b c d e f 1 1 0 0 0 1 1 2 1 1 0 0 0 0 I =   G 3 0 1 1 0 0 1 4 0 0 1 1 0 0 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 \ g

Graph minor

Let G = (V,E) be a graph. • G/e Contraction an edge e ∈ E • G \ e Deletion an edge e ∈ E • G \ v Deletion a vertex v ∈ V

f

e g

G f f

g

G/e G/e \ g

Graph minor

Let G = (V,E) be a graph. • G/e Contraction an edge e ∈ E • G \ e Deletion an edge e ∈ E • G \ v Deletion a vertex v ∈ V

f

e g

G f f

g

G/e G/e \ g

Graph minor

Let G = (V,E) be a graph. • G/e Contraction an edge e ∈ E • G \ e Deletion an edge e ∈ E • G \ v Deletion a vertex v ∈ V

f

e g

G f

G/e \ g

Graph minor

Let G = (V,E) be a graph. • G/e Contraction an edge e ∈ E • G \ e Deletion an edge e ∈ E • G \ v Deletion a vertex v ∈ V

f f

e g g

G G/e Graph minor

Let G = (V,E) be a graph. • G/e Contraction an edge e ∈ E • G \ e Deletion an edge e ∈ E • G \ v Deletion a vertex v ∈ V

f f f

e g g

G G/e G/e \ 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 ∈ 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 ∈ 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 ∈ 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 ∈ 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 \{v1, v3, v5}

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 \{v1, v3, v5}

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 \{v1, v3, v5}

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 \{v1, v3, v5}

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 \{v1, v3, v5}

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. • Natural way. • Use matroid.

Theorem

Theorem The graph B6 could not be a pivot-minor of the incidence graph of a binary tree. Theorem

Theorem The graph B6 could not be a pivot-minor of the incidence graph of a binary tree.

We have two proofs of the theorem. • Natural way. • Use matroid. Theorem

Theorem The graph B6 could not be a pivot-minor of the incidence graph of a binary tree.

We have two proofs of the theorem. • Natural way. • Use matroid. 1. First proof Sketch of proof 1 We can first every pivot, and last delete. 2 Every vertex can be appeared at most once in pivoting set. EX : G ∧ ab ∧ ac ∧ bd ∧ ce ∧ ae = G ∧ ad. XV \ X   3 If M = XAB and A is nonsingular, V \ XCD

then we define XV \ X −1 −1  M ∗ X = XA A B . V \ X −CA−1 D − CA−1B

4 A graph H is obtained from G by pivoting edges, then there is X ⊆ V (G) such that A(G)[X] is nonsingular and A(G) ∗ X = A(H) Sketch of proof 1 We can first every pivot, and last delete. 2 Every vertex can be appeared at most once in pivoting set. EX : G ∧ ab ∧ ac ∧ bd ∧ ce ∧ ae = G ∧ ad. XV \ X   3 If M = XAB and A is nonsingular, V \ XCD

then we define XV \ X −1 −1  M ∗ X = XA A B . V \ X −CA−1 D − CA−1B

4 A graph H is obtained from G by pivoting edges, then there is X ⊆ V (G) such that A(G)[X] is nonsingular and A(G) ∗ X = A(H) Sketch of proof 1 We can first every pivot, and last delete. 2 Every vertex can be appeared at most once in pivoting set. EX : G ∧ ab ∧ ac ∧ bd ∧ ce ∧ ae = G ∧ ad. XV \ X   3 If M = XAB and A is nonsingular, V \ XCD

then we define XV \ X −1 −1  M ∗ X = XA A B . V \ X −CA−1 D − CA−1B

4 A graph H is obtained from G by pivoting edges, then there is X ⊆ V (G) such that A(G)[X] is nonsingular and A(G) ∗ X = A(H) Sketch of proof 1 We can first every pivot, and last delete. 2 Every vertex can be appeared at most once in pivoting set. EX : G ∧ ab ∧ ac ∧ bd ∧ ce ∧ ae = G ∧ ad. XV \ X   3 If M = XAB and A is nonsingular, V \ XCD

then we define XV \ X −1 −1  M ∗ X = XA A B . V \ X −CA−1 D − CA−1B

4 A graph H is obtained from G by pivoting edges, then there is X ⊆ V (G) such that A(G)[X] is nonsingular and A(G) ∗ X = A(H) Sketch of proof

Theorem(Graphs and matrices..) Let T be a tree. Then A(T ) is nonsingular iff T has a perfect matching. Note. Sufficient and Necessary to connect after pivoting 1. Exist an alternating path between two vertices. 2. Both are in pivoting set → (start:∼ , end:∼) One is in pivoting set, one is in boundary → (start:-, end:∼) Both are in boundary → (start:-, end:-)

Sketch of proof

If B6 is a pivot-minor of the incidence graph of a binary tree ...

v0 v1 v2 v3 v4 vn−1 vn vn+1 Sketch of proof

If B6 is a pivot-minor of the incidence graph of a binary tree ...

v0 v1 v2 v3 v4 vn−1 vn vn+1

Note. Sufficient and Necessary to connect after pivoting 1. Exist an alternating path between two vertices. 2. Both are in pivoting set → (start:∼ , end:∼) One is in pivoting set, one is in boundary → (start:-, end:∼) Both are in boundary → (start:-, end:-) Sketch of proof

a b

B6 (X) (X)

Proof of the Theorem a or b must in a closure of X. There are 4 types of vertices. • X ∩ C(T ) • X \ C(T ) • N(X) ∩ C(T ) • N(X) \ C(T )

v0 v1 v2 v3 v4 vn−1 vn vn+1 (X) (X)

Proof of the Theorem a or b must in a closure of X. There are 4 types of vertices. • X ∩ C(T ) • X \ C(T ) • N(X) ∩ C(T ) • N(X) \ C(T )

v0 v1 v2 v3 v4 vn−1 vn vn+1 (X) (X)

Proof of the Theorem a or b must in a closure of X. There are 4 types of vertices. • X ∩ C(T ) • X \ C(T ) • N(X) ∩ C(T ) • N(X) \ C(T )

v0 v1 v2 v3 v4 vn−1 vn vn+1 (X) (X)

Proof of the Theorem a or b must in a closure of X. There are 4 types of vertices. • X ∩ C(T ) • X \ C(T ) • N(X) ∩ C(T ) • N(X) \ C(T )

v0 v1 v2 v3 v4 vn−1 vn vn+1 Proof of the Theorem a or b must in a closure of X. There are 4 types of vertices. • X ∩ C(T ) • X \ C(T ) (X) • N(X) ∩ C(T ) (X) • N(X) \ C(T )

v0 v1 v2 v3 v4 vn−1 vn vn+1 Proof of the Theorem a or b must in a closure of X. There are 4 types of vertices. • X ∩ C(T ) • X \ C(T ) (X) • N(X) ∩ C(T ) (X) • N(X) \ C(T )

v0 v1 v2 v3 v4 vn−1 vn vn+1 Proof of the Theorem

It is ok. But is there more simpler proof? 2. Second proof Matroid Minor

There are minor operations for a matroid. (Need to know dual..)

Note. For a graph G and H, if M(H) is a minor of M(G) then H is a minor of G. The fundamental graph of a matroid

B : a base in M. A bipartite graph H with bipartition B ∪ (E(M) \ B) is a fundamental graph of M with respect to B vw ∈ E(H) if and only if v ∈ B and w ∈ E(M) \ B and B \{v} ∪ {w} is a base of M.

a e a f f c G b d e b H c d

{a, f, e, d} ∈ I : a base Binary Matroid from a Bipartite Graph Let G be a bipartite graph with a bipartition A ∪ B. Bin(G, A, B) be the binary matroid on V represented by the A × V matrix (IA,A(G)[A, B]) where IA is the A × A identity matrix.

a f c e b H d

a f e d c b a 1 0 0 0 0 1 f 0 1 0 0 1 1 (IA,A(G)[A, B]) =   e 0 0 1 0 1 0 d 0 0 0 1 1 0 Binary Matroid from a Bipartite Graph Lemma Let G be a graph. Let B be a base of the matroid M(G) and H be the fundamental graph of the matroid M(G) with respect to B. Then M(G) = Bin(H,B,E(G) \ B).

a b c d e f a f e d c b 1 1 0 0 0 1 1 1 1 1 1 0 0 0 2 1 1 0 0 0 0 2 1 0 0 0 0 1 I =  =   G 3 0 1 1 0 0 1 3 0 1 0 0 1 1 4 0 0 1 1 0 0 4 0 0 0 1 1 0 5 0 0 0 1 1 0 5 0 0 1 1 0 0

a f e d c b a 1 0 0 0 0 1 f 0 1 0 0 1 1 (IA,A(G)[A, B]) =   e 0 0 1 0 1 0 d 0 0 0 1 1 0 Minor

Bin(H,A0,B0)

Theorem.

Theorem(Oum) Let G be a bipartite graph with a bipartition A ∪ B = V (G). If H is a pivot-minor of G, then there is a bipartition A0 ∪ B0 = V (H) such that Bin(H,A0,B0) is a minor of Bin(G, A, B).

G Bin(G, A, B)

Pivot minor

H Theorem.

Theorem(Oum) Let G be a bipartite graph with a bipartition A ∪ B = V (G). If H is a pivot-minor of G, then there is a bipartition A0 ∪ B0 = V (H) such that Bin(H,A0,B0) is a minor of Bin(G, A, B).

G Bin(G, A, B)

Pivot minor Minor

H Bin(H,A0,B0) Theorem.

Theorem(Oum) Let G be a bipartite graph with a bipartition A ∪ B = V (G). If H is a pivot-minor of G, then there is a bipartition A0 ∪ B0 = V (H) such that Bin(H,A0,B0) is a minor of Bin(G, A, B).

G Bin(G, A, B)

Pivot minor Minor

H Bin(H,A0,B0) Bin(P, A, B) = M(T )

Minor

0 0 Bin(B6,A ,B ) = M(?)

Theorem.

T P

P : Incid. graph of a binary tree

Pivot minor

B6 = M(T )

= M(?)

Theorem.

T P

P : Incid. graph of a binary tree Bin(P, A, B)

Pivot minor Minor

0 0 B6 Bin(B6,A ,B ) = M(?)

Theorem.

T P

P : Incid. graph of a binary tree Bin(P, A, B) = M(T )

Pivot minor Minor

0 0 B6 Bin(B6,A ,B ) Theorem.

T P

P : Incid. graph of a binary tree Bin(P, A, B) = M(T )

Pivot minor Minor

0 0 B6 Bin(B6,A ,B ) = M(?) Convert.

a b

B6 graph

Which graphs have B6 as the fundamental graph? Note. G1 and G2 have G3 as a minor.

G3

Convert.

Use Whitney’s 2-isomorphism theorem.

f h f h e g e g a b c d a d c b G1 G2 Convert.

Use Whitney’s 2-isomorphism theorem.

f h f h e g e g a b c d a d c b G1 G2

Note. G1 and G2 have G3 as a minor.

G3 Theorem.

T P

P : Incid. graph of a binary tree Bin(P, A, B) = M(T ) T

Pivot minor Minor Minor

0 0 B6 Bin(B6,A ,B ) = M(G1) G1 or M(G2) or G2 Proof

T

G3

Multiple edges in a minor of T must contain the apex vertex.

Therefore, T cannot have G3 as a minor. Done. Discussion

• Is there a parameter that solve this problem? • Is there another characterization of singularity(over GF (2)) of the adjacency matrices of graphs? • Fundamental graph is always bipartite. Is there a way to help for working on pivot-minors of non-bipartite graphs? Discussion

• Is there a parameter that solve this problem? • Is there another characterization of singularity(over GF (2)) of the adjacency matrices of graphs? • Fundamental graph is always bipartite. Is there a way to help for working on pivot-minors of non-bipartite graphs? Discussion

• Is there a parameter that solve this problem? • Is there another characterization of singularity(over GF (2)) of the adjacency matrices of graphs? • Fundamental graph is always bipartite. Is there a way to help for working on pivot-minors of non-bipartite graphs? Discussion

• Is there a parameter that solve this problem? • Is there another characterization of singularity(over GF (2)) of the adjacency matrices of graphs? • Fundamental graph is always bipartite. Is there a way to help for working on pivot-minors of non-bipartite graphs? Thank you for your attention