Introduction to Delta-Matroids

Transcription

1 Introduction to Delta-Matroids Carolyn Chun, Iain Moffatt, Steve Noble, Ralf Rueckriemen Brunel University 23/7/2014 Steve Noble ( Brunel University ) Introduction to Delta-Matroids 23/7/ / 20

2 Ribbon graphs Ribbon graphs A topological graph with discs for vertices, ribbons for edges. Considered up to homeomorphisms that preserve vertex-edge structure (including cyclic order of edges at vertices). = Topologically a punctured surface. Steve Noble ( Brunel University ) Introduction to Delta-Matroids 23/7/ / 20

3 Ribbon graphs and quasi-trees Definition A E is a quasi-tree in a connected ribbon graph (V, E) if the boundary of (V, A) has one component. Example A spanning tree is a quasi-tree. a b e c d abcd acde bcde ab ac ad ae bc bd be Steve Noble ( Brunel University ) Introduction to Delta-Matroids 23/7/ / 20

4 What is it? R? G M Steve Noble ( Brunel University ) Introduction to Delta-Matroids 23/7/ / 20

5 Delta-Matroid Definition (E, B) form a matroid if 1 B =. 2 If B 1, B 2 B and e B 1 B 2 then there exists f B 1 B 2 such that B 1 {e, f } B. 3 B 1 = B 2 for all B 1, B 2 B. Definition (Bouchet) (E, F) form a delta-matroid if 1 F. 2 If F 1, F 2 F and e F 1 F 2 then there exists f F 1 F 2 such that F 1 {e, f } F. We allow e = f in the definition. F is the set of feasible sets. Steve Noble ( Brunel University ) Introduction to Delta-Matroids 23/7/ / 20

6 Delta-matroids and ribbon graphs Theorem (Bouchet) The quasi-trees of a ribbon graph form the feasible sets of a delta-matroid. The ribbon graph can be embedded in an orientable surface if and only if the sizes of all feasible sets have the same parity. (Such delta-matroids are called even). Proposition Every matroid is a delta-matroid. Proposition (Bouchet) The feasible sets of smallest (largest) size form the bases of a matroid D min (D max ). Steve Noble ( Brunel University ) Introduction to Delta-Matroids 23/7/ / 20

7 Deletion and contraction in ribbon graphs b a a b a Big new vertex e d c b G non-loop c d c G e non-orientable loop d G/e orientable loop G G\e G/e Steve Noble ( Brunel University ) Introduction to Delta-Matroids 23/7/ / 20

8 Deletion and contraction in delta-matroids Just like matroids... Definition A coloop is an element in every feasible set. A loop is an element in no feasible set. Definition 1 If e is not a coloop in D, then F(D \ e) = {F : F F(D), e / F}. 2 If e is not a loop in D, then F(D/e) = {F e : F F(D), e F}. 3 If e is either a coloop or a loop then D/e = D \ e. Steve Noble ( Brunel University ) Introduction to Delta-Matroids 23/7/ / 20

9 Example Proposition Deletion and contraction in ribbon graphs corresponds to deletion and contraction in delta-matroids. 1 G= 2 3 D(G) D(G/1) D(G \ 1) Steve Noble ( Brunel University ) Introduction to Delta-Matroids 23/7/ / 20

10 Hiding edges Edges can be described by pairs of labelled arrows on the boundary: 1 orient edge e 2 add arrows where e meets vertices 3 remove edge. e e e e e e Example = 2 2 = = = 1 1 = = = 3 Steve Noble ( Brunel University ) Introduction to Delta-Matroids 23/7/ / 20

11 The geometric dual of a ribbon graph The geometric dual G of G One vertex of G in each face of G. One edge of G whenever faces of G are adjacent. embed dual expand G= take expand redraw ribbon graph Steve Noble ( Brunel University ) Introduction to Delta-Matroids = G 23/7/ / 20

12 The geometric dual of a ribbon graph The geometric dual G of G One vertex of G in each face of G. One edge of G whenever faces of G are adjacent. embed dual expand G= take expand redraw ribbon graph = G The geometric dual G of a ribbon graph G Fill in punctures of surface G with vertices of G, then delete vertices of G to get G. Steve Noble ( Brunel University ) Introduction to Delta-Matroids 23/7/ / 20

13 Partial duals The partial dual G A of G is obtained by forming the dual only at the edges in A E(G). Definition: partial duals (S. Chmutov 07) 1 A E(G) 2 Replace edges not in A by arrows. 3 Form geometric dual. 4 Add back hidden edges. 5 Gives the partial dual G A. Example G = e = G {e} Steve Noble ( Brunel University ) Introduction to Delta-Matroids 23/7/ / 20

14 Partial dual of a single edge G non-loop non-orientable loop orientable loop G e Steve Noble ( Brunel University ) Introduction to Delta-Matroids 23/7/ / 20

15 Another example Forming G A with A = {2, 3}. 1 G= 2 = 3 1: given G and A 2: hide edges not in A = = 3: form the dual 4 & 5: add edge back to get G A Steve Noble ( Brunel University ) Introduction to Delta-Matroids 23/7/ / 20

16 The example continued... 1 G= 2 3 has four partial duals (up to isomorphism): Observe that G and G A can have very different graph theoretic and topological properties. Steve Noble ( Brunel University ) Introduction to Delta-Matroids 23/7/ / 20

17 Partial dual Definition The partial dual or twist of the delta-matroid D = (E, F) by a set A E is the delta-matroid D A on E with feasible sets {F A : F F}. Proposition Partial duals of ribbon graphs and delta-matroids correspond. Steve Noble ( Brunel University ) Introduction to Delta-Matroids 23/7/ / 20

18 Another class of delta-matroids Partial duals of matroids form a minor-closed family of delta-matroids. Theorem (Duchamp) The excluded minors for being a partial dual of a matroid are Theorem (CMNR) D 0 D 1 D D A is a matroid if and only if A is separating in D min and both D \ A and D \ A c are matroids. Steve Noble ( Brunel University ) Introduction to Delta-Matroids 23/7/ / 20

19 Bipartite and Eulerian matroids Definition A bipartite (Eulerian) matroid is one in which all circuits (cocircuits) have an even number of elements. Theorem (CMNR) If M is a bipartite (Eulerian) binary matroid then (M A) min is bipartite (Eulerian) if and only if A is a bicycle i.e. in both the cycle and cocycle spaces. Steve Noble ( Brunel University ) Introduction to Delta-Matroids 23/7/ / 20

20 Representable delta-matroids Binary delta-matroids A symmetric binary matrix The non-singular principal sub-matrices Steve Noble ( Brunel University ) Introduction to Delta-Matroids 23/7/ / 20

21 Representable delta-matroids Binary delta-matroids A symmetric binary matrix The non-singular principal sub-matrices Definition (Bouchet) Binary delta-matroids are partial duals of those formed from symmetric binary matrices. This extends to any field with a slight technicality. Steve Noble ( Brunel University ) Introduction to Delta-Matroids 23/7/ / 20

22 Representable delta-matroids Binary delta-matroids A symmetric binary matrix The non-singular principal sub-matrices Proposition (Bouchet) Every matroid representable over F is representable over F as a delta-matroid. Steve Noble ( Brunel University ) Introduction to Delta-Matroids 23/7/ / 20

23 Questions Any questions? Steve Noble ( Brunel University ) Introduction to Delta-Matroids 23/7/ / 20

24 Questions Any questions? Over to Carolyn... Steve Noble ( Brunel University ) Introduction to Delta-Matroids 23/7/ / 20