Single-element Extensions Research Articles

Cobiased Graphs: Single-Element Extensions and Elementary Quotients of Graphic Matroids

Zaslavsky (1991) introduced a graphical structure called a biased graph and used it to characterize all single-element coextensions and elementary lifts of graphic matroids. We introduce a new graphical structure that we call a cobiased graph and use it to characterize single-element extensions and elementary quotients of graphic matroids.

Whitney numbers of matroid extensions and co-extensions

Crapo found that single-element extensions of matroid are equivalent to modular cuts. This paper will study the single-element extension and co-extension of a matroid and establish the upper semi-continuity for the signless coefficients of Whitney polynomials and Whitney numbers on both extensions via its extension lattice, a lattice of all non-empty modular cuts.

Enumeration of extensions of the cycle matroid of a complete graph

We prove that the number of single element extensions of the cycle matroid of the complete graph Kn+1 is 2(nn/2)(1+o(1)). This is done using a characterization of extensions as “linear subclasses”.

Powerful Sets: a Generalisation of Binary Matroids

A set $S\subseteq\{0,1\}^E$ of binary vectors, with positions indexed by $E$, is said to be a powerful code if, for all $X\subseteq E$, the number of vectors in $S$ that are zero in the positions indexed by $X$ is a power of 2. By treating binary vectors as characteristic vectors of subsets of $E$, we say that a set $S\subseteq2^E$ of subsets of $E$ is a powerful set if the set of characteristic vectors of sets in $S$ is a powerful code. Powerful sets (codes) include cocircuit spaces of binary matroids (equivalently, linear codes over $\mathbb{F}_2$), but much more besides. Our motivation is that, to each powerful set, there is an associated nonnegative-integer-valued rank function (by a construction of Farr), although it does not in general satisfy all the matroid rank axioms.In this paper we investigate the combinatorial properties of powerful sets. We prove fundamental results on special elements (loops, coloops, frames, near-frames, and stars), their associated types of single-element extensions, various ways of combining powerful sets to get new ones, and constructions of nonlinear powerful sets. We show that every powerful set is determined by its clutter of minimal nonzero members. Finally, we show that the number of powerful sets is doubly exponential, and hence that almost all powerful sets are nonlinear.

Unavoidable Connected Matroids Retaining a Specified Minor

A sufficiently large connected matroid $M$ contains a big circuit or a big cocircuit. Wu showed that we can ensure that $M$ has a big circuit or a big cocircuit containing any chosen element of $M$. In this paper, we prove that, for a fixed connected matroid $N$, if $M$ is a sufficiently large connected matroid having $N$ as a minor, then, up to duality, either $M$ has a big connected minor in which $N$ is a spanning restriction and the deletion of $E(N)$ is a large connected uniform matroid, or $M$ has, as a minor, the $2$-sum of a big circuit and a connected single-element extension or coextension of $N$. In addition, we find a set of unavoidable minors for the class of graphs that have a cycle and a bond with a big intersection.

On Seymour’s Decomposition Theorem

The Splitter Theorem states that, if $N$ is a 3-connected proper minor of a 3-connected matroid $M$ such that, if $N$ is a wheel or whirl then $M$ has no larger wheel or whirl, respectively, then there is a sequence $M_0,..., M_n$ of 3-connected matroids with $M_0\cong N$, $M_n=M$ and for $i\in \{1,..., n\}$, $M_i$ is a single-element extension or coextension of $M_{i-1}$. Observe that there is no condition on how many extensions may occur before a coextension must occur. In this paper, we give a strengthening of the Splitter Theorem, as a result of which we can obtain, up to isomorphism, $M$ starting with $N$ and at each step doing a 3-connected single-element extension or coextension, such that at most two consecutive single-element extensions occur in the sequence (unless the rank of the matroids involved are $r(M)$). Moreover, if two consecutive single-element extensions by elements $\{e, f\}$ are followed by a coextension by element $g$, then $\{e, f, g\}$ form a triad in the resulting matroid. Using the Strong Splitter Theorem, we make progress toward the problem of determining the almost-regular matroids [6, 15.9.8]. {\it Find all 3-connected non-regular matroids such that, for all $e$, either $M\backslash e$ or $M/e$ is regular.} In [4] we determined the binary almost-regular matroids with at least one regular element (an element such that both $M\backslash e$ and $M/e$ is regular) by characterizing the class of binary almost-regular matroids with no minor isomorphic to one particular matroid that we called $E_5$. As a consequence of the Strong Splitter Theorem we can determine the class of binary matroids with an $E_5$-minor, but no $E_4$-minor.

Enumeration of 2-Polymatroids on up to Seven Elements

A theory of single-element extensions of integer polymatroids analogous to that of matroids is developed. We present an algorithm to generate a catalog of 2-polymatroids, up to isomorphism. When we implemented this algorithm on a computer, obtaining all 2-polymatroids on at most seven elements, we discovered the surprising fact that the number of 2-polymatroids on seven elements fails to be unimodal in rank.

Strong Splitter Theorem

The Splitter Theorem states that, if N is a 3-connected proper minor of a 3-connected matroid M such that, if N is a wheel or whirl then M has no larger wheel or whirl, respectively, then there is a sequence M 0, . . . , M n of 3-connected matroids with \({M_0 \cong N}\), M n = M and for \({i \in \{1, \ldots , n}\}\), M i is a single-element extension or coextension of M i−1. Observe that there is no condition on how many extensions may occur before a coextension must occur. We give a strengthening of the Splitter Theorem, as a result of which we can obtain, up to isomorphism, M starting with N and at each step doing a 3-connected single-element extension or coextension, such that at most two consecutive single-element extensions occur in the sequence (unless the rank of thematroids involved is r(M)). Moreover, if two consecutive single-element extensions by elements {e, f} are followed by a coextension by element g, then {e, f , g} form a triad in the resulting matroid.

Capturing matroid elements in unavoidable 3-connected minors

A result of Ding, Oporowski, Oxley, and Vertigan reveals that a large 3-connected matroid M has unavoidable structure. For every n>2, there is an integer f(n) so that if |E(M)|>f(n), then M has a minor isomorphic to the rank-n wheel or whirl, a rank-n spike, the cycle or bond matroid of K3,n, or U2,n or Un−2,n. In this paper, we build on this result to determine what can be said about a large structure using a specified element e of M. In particular, we prove that, for every integer n exceeding two, there is an integer g(n) so that if |E(M)|>g(n), then e is an element of a minor of M isomorphic to the rank-n wheel or whirl, a rank-n spike, the cycle or bond matroid of K1,1,1,n, a specific single-element extension of M(K3,n) or the dual of this extension, or U2,n or Un−2,n.

Restrictions as stabilizers

Two GF ( q ) -representations of a matroid are projectively equivalent if one can be obtained from the other by elementary row operations or column scaling. If, in addition, we allow field automorphisms, then the representations are equivalent. A matroid stabilizes its single-element extensions over GF ( q ) , if no two GF ( q ) -representations of the matroid can be extended to two projectively inequivalent GF ( q ) -representations of its extension. We prove that if a 3-connected GF ( q ) -representable matroid stabilizes its single-element extensions, then it stabilizes any GF ( q ) -representable matroid that contains it as a restriction. As a result, a GF ( q ) -representable matroid with a line containing at least q points is stabilized by that line and a GF ( q ) -representable matroid with a plane containing at least 2 q points, but no line containing at least q points, is stabilized by that plane.

Amalgams of extremal matroids with no [formula omitted]-minor

For an integer ℓ ≥ 2 , let U ( ℓ ) be the class of matroids with no U 2 , ℓ + 2 -minor. A matroid in U ( ℓ ) is extremal if it is simple and has no simple rank-preserving single-element extension in U ( ℓ ) . An amalgam of two matroids is a simultaneous extension of both on the union of the two ground sets. We study amalgams of extremal matroids in U ( ℓ ) : we determine which amalgams are in U ( ℓ ) and which are extremal in U ( ℓ ) .

Internally 4-connected binary matroids with cyclically sequential orderings

We characterize all internally 4-connected binary matroids M with the property that the ground set of M can be ordered ( e 0 , … , e n − 1 ) in such a way that { e i , … , e i + t } is 4-separating for all 0 ≤ i , t ≤ n − 1 (all subscripts are read modulo n ). We prove that in this case either n ≤ 7 or, up to duality, M is isomorphic to the polygon matroid of a cubic or quartic planar ladder, the polygon matroid of a cubic or quartic Möbius ladder, a particular single-element extension of a wheel, or a particular single-element extension of the bond matroid of a cubic ladder.

A single-element extension of antimatroids

An antimatroid is a family of sets which is accessible, closed under union, and includes an empty set. A number of examples of antimatroids arise from various kinds of shellings and searches on combinatorial objects, such as, edge/node shelling of trees, poset shelling, node-search on graphs, etc. (Discrete Math. 78 (1989) 223; Geom. Dedicata 19 (1985) 247; Greedoids, Springer, Berlin, 1980) [1–3]. We introduce a one-element extension of antimatroids, called a lifting, and the converse operation, called a reduction. It is shown that a family of sets is an antimatroid if and only if it is constructed by applying lifting repeatedly to a trivial lattice. Furthermore, we introduce two specific types of liftings, 1-lifting and 2-lifting, and show that a family of sets is an antimatroid of poset shelling if and only if it is constructed from a trivial lattice by repeating 1-lifting. Similarly, an antimatroid of edge-shelling of a tree is shown to be constructed by repeating 2-lifting, and vice versa.

Generation of Oriented Matroids—A Graph Theoretical Approach

We discuss methods for the generation of oriented matroids and of isomorphism classes of oriented matroids. Our methods are based on single element extensions and graph theoretical representations of oriented matroids, and all these methods work in general rank and for non-uniform and uniform oriented matroids as well. We consider two types of graphs, cocircuit graphs and tope graphs, and discuss the single element extensions in terms of localizations which can be viewed as partitions of the vertex sets of the graphs. Whereas localizations of the cocircuit graph are well characterized, there is no graph theoretical characterization known for localizations of the tope graph. In this paper we prove a connectedness property for tope graph localizations and use this for the design of algorithms for the generation of single element extensions by use of tope graphs. Furthermore, we discuss similar algorithms which use the cocircuit graph. The characterization of localizations of cocircuit graphs finally leads to a backtracking algorithm which is a simple and efficient method for the generation of single element extensions. We compare this method with a recent algorithm of Bokowski and Guedes de Oliveira for uniform oriented matroids.

Stabilizers of Classes of Representable Matroids

Let M be a class of matroids representable over a field F . A matroid N ∈ M stabilizes M if, for any 3-connected matroid M ∈ M , an F -representation of M is uniquely determined by a representation of any one of its N -minors. One of the main theorems of this paper proves that if M is minor-closed and closed under duals, and N is 3-connected, then to show that N is a stabilizer it suffices to check 3-connected matroids in M that are single-element extensions or coextensions of N , or are obtained by a single-element extension followed by a single-element coextension. This result is used to prove that a 3-connected quaternary matroid with no U 3, 6 -minor has at most ( q −2)( q −3) inequivalent representations over the finite field GF ( q ). New proofs of theorems bounding the number of inequivalent representations of certain classes of matroids are given. The theorem on stabilizers is a consequence of results on 3-connected matroids. It is shown that if N is a 3-connected minor of the 3-connected matroid M , and | E ( M )− E ( N )|⩾3, then either there is a pair of elements x , y ∈ E ( M ) such that the simplifications of M / x , M / y , and M / x , y are all 3-connected with N -minors or the cosimplifications of M \ x , M \ y , and M \ x , y are all 3-connected with N -minors, or it is possible to perform a Δ − Y or Y − Δ exchange to obtain a matroid with one of the above properties.

The Hadamard matroid and an anomaly in its single element extensions

A nonstandard vector space is formulated, whose bases afford a representation of what is called a Hadamard matroid, M p . For prime p, existence of M p is equivalent to the existence of both a classical Hadamard matrix H( p, p) and a certain affine resolvable, balanced incomplete block design AR( p). An anomaly in the representable single element extension of a Hadamard matroid is discussed.

Steiner Complexes, Matroid Ports, and Shellability

We study the theory of matroid Steiner families, which were introduced in a recent paper of Colbourri and Pulleyblank as abstractions of Steiner trees on graphs. Our primary result is that these families are equivalent to matroid ports.Using this equivalence, we relate these Steiner families to two fundamental matroid-theoretic constructions namely, single-element extensions and elementary quotients. We also discuss a new and very general class of Steiner families, which arise as minimal supports of affine systems. The latter half of the paper deals with the structure of Steiner complexes, which are independence systems associated with these Steiner families. We develop the notion of an S-partition of a simplicial complex which extends the notion of a shelling of a pure simplicial complex. We develop an S-partitioning theory for Steiner complexes generalizing the well-known shellability theory for the collection of independent sets of a matroid.

Sur la compatibilité des extensions ponctuelles d'un matroïde

A natural sufficient condition for a finite family of single element extensions of a matroid to be compatible is given. Characterizations of all the finite extensions N of a matroid M( E) are given for which the rank function satisfies ρ N(X)= Min Z⊂E {ρ M(Z)+|X− Z N|} or equivalently the closure operator satisfies X N = X N ⌢ E N ⌣ X . The single element extensions and the principal extensions are examples of such matroids. The notion of a sheaf of flats of M. Las Vergnas is used in the proof of a new necessary and sufficient condition for two single element extensions of a matroid to be compatible. An initial announcement of part of these results appeared in R. Cordovil ( C. R. Acad. Sci. Paris. A 284 (1977), 1249–1252).

Majors of geometric strong maps

Most of the constructions in the theory of combinatorial geometries take place in the category of pregeometries and strong maps. In this present paper, we study these constructions and the structure of pregeometries by factoring strong maps into elementary maps, after the work of Dowling and Kelly. Using the modular cuts of the factorization to determine certain single-element extensions, we associate each factorization Φ to a unique labeled pregeometry, called the major of Φ. Every major of a factorization of a strong map f:H→G has H and G as distinguished minors: H as a subgeometry; G as a contraction.A partial order is defined on the set of all factorizations of f, ▪, and a compatible partial order is defined on the set of all majors of f, ▪. The map Φ : ▪→▪ is shown to be an increasing surjection, while the map Y : ▪→ ▪ is shown to be an increasing injection. The composition Φ ∘ Y is shown to be the identity on ▪, while Y ∘ Φ is a decreasing map on ▪, defining a co-closure on majors. The partial order on ▪, although seemingly more restrictive than necessary, is shown to be necessary to reflect the order on factorizations.Special consideration is given the zero element of ▪, called the Higgs factorization of f, which is constructed using the lift construction of Higgs. We explore the connection between the majors of this theory and a particular major constructed by Higgs, and show that this Higgs major is the major of the Higgs factorization. As the unique smallest major of any strong map, the Higgs major thus defines a unary operator on pregeometries: Y(G) is the Higgs major of f : ▪→G.

Modular constructions for combinatorial geometries

R. Stanley, in an investigation of modular flats in geometries (Algebra Universalis 1-2 (1971), 214-217), proved that the characteristic polynomial χ ( x ) \chi (x) of a modular flat x x divides the characteristic polynomial χ ( G ) \chi (G) of a geometry G G . In this paper we identify the quotient: THEOREM. If x x is a modular flat of G , χ ( G ) / χ ( x ) = χ ( T x ¯ ( G ) ) / ( λ − 1 ) G,\chi (G)/\chi (x) = \chi (\overline {{T_x}} (G))/(\lambda - 1) , where T x ¯ ( G ) \overline {{T_x}} (G) is the complete Brown truncation of G G by x x . (The lattice of T x ¯ ( G ) \overline {{T_x}} (G) consists of all flats containing x x and all flats disjoint from x x , with the induced order from G G .) We give many characterizations of modular flats in terms of their lattice properties as well as by means of a short-circuit axiom and a modular version of the MacLane-Steinitz exchange axiom. Modular flats are shown to have many of the useful properties of points and distributive flats (separators) in addition to being much more prevalent. The theorem relating the chromatic polynomials of two graphs and the polynomial of their vertex join across a common clique generalizes to geometries: THEOREM. Given geometries G G and H H , if x x is a modular flat of G G as well as a subgeometry of H H , then there exists a geometry P = P x ( G , H ) P = {P_x}(G,H) which is a pushout in the category of injective strong maps and such that χ ( P ) = χ ( G ) χ ( H ) / χ ( x ) \chi (P) = \chi (G)\chi (H)/\chi (x) . The closed set structure, rank function, independent sets, and lattice properties of P P are characterized. After proving a modular extension theorem we give applications of our results to Crapo’s single element extension theorem, Crapo’s join operation, chain groups, unimodular geometries, transversal geometries, and graphs.