Introduction to multimatroids and their polynomials Robert Brijder - - PowerPoint PPT Presentation
Introduction to multimatroids and their polynomials Robert Brijder Hasselt University, Belgium Dagstuhl, June 12-17, 2016 Robert Brijder Multimatroid introduction Motivation: 4-regular graphs v 2 e 1 t v , 1 = {{ e 1 , e 4 } , { e 2 , e 3 }}
Robert Brijder
Hasselt University, Belgium
Dagstuhl, June 12-17, 2016
Robert Brijder Multimatroid introduction
G v1 v2 v3 v4
v e1 e2 e3 e4
v′ v′′ v′ v′′ v′ v′′
G|tv,1 G|tv,2 G|tv,3
tv,1 = {{e1, e4}, {e2, e3}} tv,2 = {{e1, e2}, {e3, e4}} tv,3 = {{e1, e3}, {e2, e4}}
Robert Brijder Multimatroid introduction
Triple ZG = (U, Ω, C), where U = all transitions = {tv,i | v ∈ V (G), i ∈ {1, 2, 3}} Ω = partition of U w.r.t. vertices = {{tv,1, tv,2, tv,3} | v ∈ V (G)} C = minimal sets of (nonconflicting) transitions such that cutting along them increases the number of connected components
G v1 v2 v3 v4
C = {{tv4,2}, {tv1,3, tv2,3}, {tv1,3, tv3,3}, {tv2,3, tv3,3}, {tv1,2, tv2,1, tv3,1}}
Robert Brijder Multimatroid introduction
Triple ZG = (U, Ω, C), where U = all transitions = {tv,i | v ∈ V (G), i ∈ {1, 2, 3}} Ω = partition of U w.r.t. vertices = {{tv,1, tv,2, tv,3} | v ∈ V (G)} C = minimal sets of (nonconflicting) transitions such that cutting along them increases the number of connected components
G v1 v2 v3 v′
4
v′′
4
C = {{tv4,2}, {tv1,3, tv2,3}, {tv1,3, tv3,3}, {tv2,3, tv3,3}, {tv1,2, tv2,1, tv3,1}}
Robert Brijder Multimatroid introduction
Triple ZG = (U, Ω, C), where U = all transitions = {tv,i | v ∈ V (G), i ∈ {1, 2, 3}} Ω = partition of U w.r.t. vertices = {{tv,1, tv,2, tv,3} | v ∈ V (G)} C = minimal sets of (nonconflicting) transitions such that cutting along them increases the number of connected components
G v1 v2 v3 v4
C = {{tv4,2}, {tv1,3, tv2,3}, {tv1,3, tv3,3}, {tv2,3, tv3,3}, {tv1,2, tv2,1, tv3,1}}
Robert Brijder Multimatroid introduction
Triple ZG = (U, Ω, C), where U = all transitions = {tv,i | v ∈ V (G), i ∈ {1, 2, 3}} Ω = partition of U w.r.t. vertices = {{tv,1, tv,2, tv,3} | v ∈ V (G)} C = minimal sets of (nonconflicting) transitions such that cutting along them increases the number of connected components
G v′
1
v′
2
v′′
1
v′′
2
v3 v4
C = {{tv4,2}, {tv1,3, tv2,3}, {tv1,3, tv3,3}, {tv2,3, tv3,3}, {tv1,2, tv2,1, tv3,1}}
Robert Brijder Multimatroid introduction
Triple ZG = (U, Ω, C), where U = all transitions = {tv,i | v ∈ V (G), i ∈ {1, 2, 3}} Ω = partition of U w.r.t. vertices = {{tv,1, tv,2, tv,3} | v ∈ V (G)} C = minimal sets of (nonconflicting) transitions such that cutting along them increases the number of connected components
G v1 v2 v3 v4
C = {{tv4,2}, {tv1,3, tv2,3}, {tv1,3, tv3,3}, {tv2,3, tv3,3}, {tv1,2, tv2,1, tv3,1}}
Robert Brijder Multimatroid introduction
Definition Let Ω be a partition of a finite set U. The elements ω ∈ Ω are called skew classes. A transversal T ⊆ U of Ω is such that |T ∩ ω| = 1 for all ω ∈ Ω. Theorem (Bouchet, 1997) For each transversal T, ZG[T] := (T, C ∩ 2T) is a matroid described by its circuits. Example
C = {{tv4,2}, {tv1,3, tv2,3}, {tv1,3, tv3,3}, {tv2,3, tv3,3}, {tv1,2, tv2,1, tv3,1}}
Transversal T = {tv1,3, tv2,3, tv3,3, tv4,2}. ZG[T] = (T, {{tv4,2}, {tv1,3, tv2,3}, {tv1,3, tv3,3}, {tv2,3, tv3,3}}) is a matroid (isomorphic to U1,3 ⊕ {loop}).
Robert Brijder Multimatroid introduction
There is at most one way to cut a vertex to increase the number of connected components.
v
v′ v′′ NO SPLIT v′ v′′ POSSIBLE SPLIT v′ v′′ NO SPLIT
G|tv,1 G|tv,2 G|tv,3
Robert Brijder Multimatroid introduction
A subtransversal of Ω is a subset of a transversal of Ω. Definition (Bouchet, 1997) Let Ω be a partition of a finite set U, and let C be a set of subtransversals of Ω. Then Z = (U, Ω, C) is called a multimatroid if
1 for all transversals T, Z[T] := (T, C ∩ 2T) is a matroid, and 2 for all C1, C2 ∈ C, there are zero or at least two ω ∈ Ω with
|(C1 ∪ C2) ∩ ω| = 2. Called “multimatroid” because it contains multiple matroids (one for each transversal). Condition 2 formalizes that there is at most one way to cut a vertex to increase the number of connected components. Multimatroids can also be defined in terms of independent sets, rank, etc. ZG is a multimatroid, where |ω| = 3 for all ω ∈ Ω.
Robert Brijder Multimatroid introduction
Definition (minor operations) Let Z = (U, Ω, C) be a multimatroid and ω ∈ Ω. For u ∈ ω, we define Z|u = (U \ ω, Ω \ {ω}, C′), where C′ =
T C(Z[T ∪ u]/u) and T ranges over all transversals
Theorem (Bouchet, 1998) Z|u is a multimatroid. Extendable to subtransversals S, denoted Z|S. Multimatroids of the form Z|S are called minors of Z. Theorem (Bouchet, 1998) For any 4-regular graph G and transition t of G, ZG|t = ZG|t.
Robert Brijder Multimatroid introduction
Theorem (Bouchet, 1997) Let G be a 4-regular graph. For all transversals T, n(ZG[T]) = c(G|T) − c(G), where n denotes the nullity of a matroid. The bases of ZG correspond 1-to-1 to Eulerian circuits of G.
Robert Brijder Multimatroid introduction
Definition Multimatroid Z is called a k-matroid if |ω| = k for all ω ∈ Ω. Example: ZG is a 3-matroid. A k-matroid has k kinds of minor operations. A 1-matroid corresponds to an ordinary matroid, but with
A 2-matroid corresponds to a delta-matroid, with both deletion and contraction. So, a matroid can be viewed as a 1-matroid and a 2-matroid.
Robert Brijder Multimatroid introduction
Definition (Bouchet, 2001) A multimatroid is tight if each minor with |Ω| = 1 has a circuit. ZG is a tight 3-matroid. Matroids form a subclass of tight 2-matroids.
v G ′ = G|S
v′ v′′ NO SPLIT v′ v′′ SPLIT! v′ v′′ NO SPLIT
G ′|tv,1 G ′|tv,2 G ′|tv,3
{tv,2} circuit of ZG ′
Robert Brijder Multimatroid introduction
Theorem Every k-matroid Z has (up-to-isomorphism) at most one tight (k + 1)-matroid Z ′ with Z = Z ′[U \ T] for some transversal T of Z ′. So, some 2-matroids can also be viewed as tight 3-matroids. This includes all quaternary matroids. So, a quaternary matroid may be viewed as a 1-matroid, a 2-matroid, and a tight 3-matroid!
Robert Brijder Multimatroid introduction
Definition (B´
enard, Bouchet, Duchamp, 1997, B, Hoogeboom, 2014)
Let Z = (U, Ω, C) be a multimatroid. We define the transition polynomial of Z as Q(Z; y) =
yn(Z[T]), where n(Z[T]) denotes the nullity of matroid Z[T]. It turns out that (a multivariate version of) Q(Z; y) subsumes various known polynomials, including: Tutte polynomial for matroids for part of the (x, y)-plane, Martin polynomial for 4-regular graphs and 2-in, 2-out graphs, interlace polynomial for graphs, and Bollob´ as-Riordan polynomial (and others) for embedded graphs [Chun, Moffatt, Noble, Rueckriemen, 2014].
Robert Brijder Multimatroid introduction
Call u ∈ U singular if {u} ∈ C. Call ω ∈ Ω singular if ω contains a singular element. Theorem Let Z be a multimatroid. If Z is the empty multimatroid, then Q(Z; y) = 1. Otherwise let ω ∈ Ω. If ω is nonsingular in Z, then Q(Z; y) =
Q(Z|v; y). If ω is singular in Z, then for all w ∈ ω Q(Z; y) = (y + |ω| − 1) Q(Z|w; y), where u ∈ ω is singular in Z.
Robert Brijder Multimatroid introduction
Theorem Let Z be a tight k-matroid with k > 1 and U = ∅. Then Q(Z; 1 − k) = 0 and For all transversals T, Q(Z[U \ T]; 1 − k) = (−1)|Ω|(1 − k)n(Z[T]).
[B, Hoogeboom, 2014]
Tutte polynomial T(M; x, y). Corollary Let M be a matroid. Case k = 2. T(M; 0, 0) = 0 and Case k = 3. If M is quaternary, then T(M; −1, −1) = (−1)|E(M)|(−2)n(B(M)), where B(M) is the “bicycle matroid” of M.
[Vertigan, 1998]
Robert Brijder Multimatroid introduction
Open problem: a useful matroid-theoretic formalization of the Martin polynomial for Eulerian graphs in general. Including, e.g., 6-regular graphs. Problem: cutting a vertex of a 6-regular graph may increase the number of connected components by two. So, perhaps Z[T] should be a polymatroid instead of a matroid? In other words, do we need a “poly-multimatroid”? Or do we cut a degree 6 vertex first into degree 4 and degree 2?
Robert Brijder Multimatroid introduction
Robert Brijder Multimatroid introduction
enard, A. Bouchet, and A. Duchamp. On the Martin and Tutte polynomials. Unpublished manuscript, January 1997.
Multimatroids I. Coverings by independent sets. SIAM Journal on Discrete Mathematics, 10(4):626–646, 1997.
Multimatroids II. Orthogonality, minors and connectivity. The Electronic Journal of Combinatorics, 5, 1998.
Multimatroids III. Tightness and fundamental graphs. European Journal of Combinatorics, 22(5):657–677, 2001.
Interlace polynomials for multimatroids and delta-matroids. European Journal of Combinatorics, 40:142–167, 2014.
Bicycle dimension and special points of the Tutte polynomial. Journal of Combinatorial Theory, Series B, 74(2):378–396, 1998.
Robert Brijder Multimatroid introduction