The highly connected even-cycle and even-cut matroids

The classes of even-cycle matroids, even-cycle matroids with a blocking pair, and even-cut matroids each have hundreds of excluded minors. We show that the number of excluded minors for these classes can be drastically reduced if we consider in each class only the highly connected matroids of sufficient size.Comment: Version 2 is a major revision, including a correction of an error in the statement of one of the main results and improved exposition. It is 89 pages, including a 33-page Jupyter notebook that contains SageMath code and that is also available in the ancillary file

The excluded 3-minors for vf-safe delta-matroids

Vf-safe delta-matroids have the desirable property of behaving well under certain duality operations. Several important classes of delta-matroids are known to be vf-safe, including the class of ribbon-graphic delta-matroids, which is related to the class of ribbon graphs or embedded graphs in the same way that graphic matroids correspond to graphs. In this paper, we characterize vf-safe delta-matroids and ribbon-graphic deltamatroids by finding the minimal obstructions, called excluded 3-minors, to membership in the class. We find the unique (up to twisted duality) excluded 3-minor within the class of set systems for the class of vf-safe delta-matroids. In the literature, binary delta-matroids appear in many different guises, with appropriate notions of minor operations equivalent to that of 3-minors, perhaps most notably as graphs with vertex minors. We give a direct explanation of this equivalence and show that some well-known results may be expressed in terms of 3-minors

There are only a finite number of excluded minors for the class of bicircular matroids

We show that the class of bicircular matroids has only a finite number of excluded minors. Key tools used in our proof include representations of matroids by biased graphs and the recently introduced class of quasi-graphic matroids. We show that if $N$ is an excluded minor of rank at least ten, then $N$ is quasi-graphic. Several small excluded minors are quasi-graphic. Using biased-graphic representations, we find that $N$ already contains one of these. We also provide an upper bound, in terms of rank, on the number of elements in an excluded minor, so the result follows.Comment: Added an appendix describing all known excluded minors. Added Gordon Royle as author. Some proofs revised and correcte

A Characterization of Certain Excluded-Minor Classes of Matroids

A result of Walton and the author establishes that every 3-connected matroid of rank and corank at least three has one of five six-element rank-3 self-dual matroids as a minor. This paper characterizes two classes of matroids that arise when one excludes as minors three of these five matroids. One of these results extends the author\u27s characterization of the ternary matroids with no M(K4)-minor, while the other extends Tutte\u27s excluded-minor characterization of binary matroids. © 1989, Academic Press Limited. All rights reserved

Minors for alternating dimaps

We develop a theory of minors for alternating dimaps --- orientably embedded digraphs where, at each vertex, the incident edges (taken in the order given by the embedding) are directed alternately into, and out of, the vertex. We show that they are related by the triality relation of Tutte. They do not commute in general, though do in many circumstances, and we characterise the situations where they do. The relationship with triality is reminiscent of similar relationships for binary functions, due to the author, so we characterise those alternating dimaps which correspond to binary functions. We give a characterisation of alternating dimaps of at most a given genus, using a finite set of excluded minors. We also use the minor operations to define simple Tutte invariants for alternating dimaps and characterise them. We establish a connection with the Tutte polynomial, and pose the problem of characterising universal Tutte-like invariants for alternating dimaps based on these minor operations.Comment: 51 pages, 7 figure

Rank connectivity and pivot-minors of graphs

The cut-rank of a set $X$ in a graph $G$ is the rank of the $X\times (V(G)-X)$ submatrix of the adjacency matrix over the binary field. A split is a partition of the vertex set into two sets $(X,Y)$ such that the cut-rank of $X$ is less than $2$ and both $X$ and $Y$ have at least two vertices. A graph is prime (with respect to the split decomposition) if it is connected and has no splits. A graph $G$ is $k^{+\ell}$-rank-connected if for every set $X$ of vertices with the cut-rank less than $k$, $\lvert X\rvert$ or $\lvert V(G)-X\rvert$ is less than $k+\ell$. We prove that every prime $3^{+2}$-rank-connected graph $G$ with at least $10$ vertices has a prime $3^{+3}$-rank-connected pivot-minor $H$ such that $\lvert V(H)\rvert =\lvert V(G)\rvert -1$. As a corollary, we show that every excluded pivot-minor for the class of graphs of rank-width at most $k$ has at most $(3.5 \cdot 6^{k}-1)/5$ vertices for $k\ge 2$. We also show that the excluded pivot-minors for the class of graphs of rank-width at most $2$ have at most $16$ vertices.Comment: 19 pages; Lemma 5.3 is now fixe

Polymatroid greedoids

AbstractThis paper discusses polymatroid greedoids, a superclass of them, called local poset greedoids, and their relations to other subclasses of greedoids. Polymatroid greedoids combine in a certain sense the different relaxation concepts of matroids as polymatroids and as greedoids. Some characterization results are given especially for local poset greedoids via excluded minors. General construction principles for intersection of matroids and polymatroid greedoids with shelling structures are given. Furthermore, relations among many subclasses of greedoids which are known so far, are demonstrated

On Local Equivalence, Surface Code States and Matroids

Recently, Ji et al disproved the LU-LC conjecture and showed that the local unitary and local Clifford equivalence classes of the stabilizer states are not always the same. Despite the fact this settles the LU-LC conjecture, a sufficient condition for stabilizer states that violate the LU-LC conjecture is missing. In this paper, we investigate further the properties of stabilizer states with respect to local equivalence. Our first result shows that there exist infinitely many stabilizer states which violate the LU-LC conjecture. In particular, we show that for all numbers of qubits $n\geq 28$, there exist distance two stabilizer states which are counterexamples to the LU-LC conjecture. We prove that for all odd $n\geq 195$, there exist stabilizer states with distance greater than two which are LU equivalent but not LC equivalent. Two important classes of stabilizer states that are of great interest in quantum computation are the cluster states and stabilizer states of the surface codes. To date, the status of these states with respect to the LU-LC conjecture was not studied. We show that, under some minimal restrictions, both these classes of states preclude any counterexamples. In this context, we also show that the associated surface codes do not have any encoded non-Clifford transversal gates. We characterize the CSS surface code states in terms of a class of minor closed binary matroids. In addition to making connection with an important open problem in binary matroid theory, this characterization does in some cases provide an efficient test for CSS states that are not counterexamples.Comment: LaTeX, 13 pages; Revised introduction, minor changes and corrections mainly in section V