The structure of delta-matroids with width one twists

TL;DR

This paper characterizes delta-matroids with a twist of width one, providing a structural theorem and an excluded minor characterization to better understand their properties and classifications.

Contribution

It introduces a Rough Structure Theorem for delta-matroids with width-one twists and offers an excluded minor characterization for such delta-matroids.

Findings

Structural theorem for delta-matroids with width-one twists

Excluded minor characterization of these delta-matroids

Enhanced understanding of delta-matroid classifications

Abstract

The width of a delta-matroid is the difference in size between a maximal and minimal feasible set. We give a Rough Structure Theorem for delta-matroids that admit a twist of width one. We apply this theorem to give an excluded minor characterisation of delta-matroids that admit a twist of width at most one.

Equations

$$\mathcal{F}(D_1)=\{\varnothing ,\{a\},\{b\},\{a,b\}\}.$$

$$\mathcal{F}(D_2)$$

$$\mathcal{F}(D_3)$$

$$\mathcal{F}(D_4)$$

$$\mathcal{F}(D_5)$$

$$\rho (A):=|E|-\min \{|A△F|:F\in \mathcal{F}\}.$$

$$w(D\ast A)=w(D|A)+w(D|\overline{A})+2{\lambda }_{D_{\min }}(A).$$

$$w(D|A)$$

$$=\rho (A)-r(A)-|E|+r(E)+|A|-r(A)+\rho (\overline{A})-r(\overline{A})-|E|+r(E)+|\overline{A}|-r(\overline{A})$$

$$=\rho (\overline{A})-|E|+\rho (A)-2(r(A)+r(\overline{A})-r(E))$$

$$=w(D\ast A)-2({\lambda }_{D_{\min }}(A)),$$

$$\{H:H\text{ is a minor of }D\ast A\}=\{J\ast (A\cap E(J)):J\text{ is a minor of }D\}.$$

$$L:=\{x\in E(D):\{x\}\in \mathcal{F}(D)\}\text{and}\overline{L}=E(D)-L.$$

$$D|\overline{A}\cong U_{0,|\overline{A}|},$$

$$\{\varnothing ,\{x\},\{y\},\{z\},\{x,y,z\}\},\{\varnothing ,\{x\},\{y\},\{x,y,z\}\},\{\varnothing ,\{x\},\{x,y,z\}\}.$$

$$\{\varnothing ,\{\alpha \},\{x,\alpha \},\{y,\alpha \},\{x,y\}\}\text{or}\{\varnothing ,\{\alpha \},\{x,\alpha \},\{y,\alpha \},\{x,y\},\{\alpha ,x,y\}\}.$$

$$\varnothing ,\{\alpha \},\{\beta \},\{x,\alpha \},\{y,\beta \},\{x,y\}.$$

$$\{\alpha ,x,y\},\{\beta ,x,y\},\{\alpha ,\beta ,x\},\{\alpha ,\beta ,y\},$$

$$\varnothing ,\{x_1,x_2\},\{x_2,x_3\},\dots ,\{x_n,x_1\}$$

$$\{x_i,x_j,x_k\}\notin \mathcal{F}(D|\{x_1,\dots ,x_n\}),\text{ for any distinct }1\le i,j,k\le n.$$

$$\{x_i,x_{i+1},x_j,x_{j+1}\}\in \mathcal{F}(D|\{x_1,\dots ,x_n\}),$$

$$\varnothing ,\{\alpha \},\{\beta \},\{\alpha ,x_2\},\{x_2,x_3\},\{x_3,x_4\},\dots ,\{x_{n-1},x_n\},\{\beta ,x_n\}.$$

$$\{x_i,x_j,x_k\}\notin \mathcal{F}(D|\{\alpha ,\beta ,x_2,\dots ,x_n\}),\text{ for any distinct }2\le i,j,k\le n.$$

$$\{\alpha ,x_{n-1},x_n\},\{\beta ,x_{n-1},x_n\}\in \mathcal{F}(D|\{\alpha ,\beta ,x_2,\dots ,x_n\}).$$

$$\{\alpha ,x_2,x_{n-1},x_n\},\{x_i,x_{i+1},x_{n-1},x_n\},\{\beta ,x_{n-2},x_{n-1},x_n\}\in \mathcal{F}(D|\{\alpha ,\beta ,x_2,\dots ,x_n\}).$$

Full text

1. Introduction, results and notation

Delta-matroids are a generalisation of matroids introduced by A. Bouchet in [1]. They can be thought of as generalising topological graph theory in the same way that matroids can be thought of as generalising graph theory (see, e.g., [4]). Roughly speaking, delta-matroids arise by dropping the requirement that bases are of the same size in the standard definition of a matroid in terms of its bases. (Formal definitions are provided below.) In the context of delta-matroids these generalised “bases” are called “feasible sets”. A basic parameter of a delta-matroid is its “width”, which is the difference between the sizes of a largest and a smallest of its feasible sets. One of the most fundamental operations in delta-matroid theory is the “twist”. In this paper we examine how the structure of a delta-matroid determines the width of the delta-matroids that are in its equivalence class under twists.

Formally, a delta-matroid $D=(E,\mathcal{F})$ consists of a finite set $E$ and a non-empty set $\mathcal{F}$ of subsets of $E$ that satisfies the Symmetric Exchange Axiom: for all $X,Y\in\mathcal{F}$, if there is an element $u\in X\triangle Y$, then there is an element $v\in X\triangle Y$ such that $X\triangle\{u,v\}\in\mathcal{F}$. Here $X\triangle Y$ denotes the symmetric difference of sets $X$ and $Y$. Note that it may be the case that $u=v$ in the Symmetric Exchange Axiom. Elements of $\mathcal{F}$ are called feasible sets and $E$ is the ground set. We often use $\mathcal{F}(D)$ and $E(D)$ to denote the set of feasible sets and the ground set, respectively, of $D$. A matroid is a delta-matroid whose feasible sets are all of the same size. In this case the feasible sets are called bases. This definition of a matroid is a straightforward reformulation of the standard one in terms of bases.

In general a delta-matroid has feasible sets of different sizes. The width of a delta-matroid, denoted $w(D)$, is the difference between the sizes of its largest and smallest feasible sets: $w(D):=\max\limits_{F\in\mathcal{F}}|F|-\min\limits_{F\in\mathcal{F}}|F|$.

Twists, introduced by Bouchet in [1], are one of the fundamental operations of delta-matroid theory. Given a delta-matroid $D=(E,{\mathcal{F}})$ and some subset $A\subseteq E$, the twist of $D$ with respect to $A$, denoted by $D*A$, is the delta-matroid given by $(E,\{A\bigtriangleup F:F\in\mathcal{F}\})$. (At times we write $D\ast e$ for $D\ast\{e\}$.) Note that the “empty twist” is $D*\emptyset=D$. The dual of $D$, written $D^{*}$, is equal to $D*E$. Moreover, in general, the twist can be thought of as a “partial dual” operation on delta-matroids.

Forming the twist of a delta-matroid usually changes the sizes of its feasible sets and its width. Here we are interested in the problem of recognising when a delta-matroid has a twist of small width. Our results are a Rough Structure Theorem for delta-matroids that have a twist of width one, and an excluded minor characterisation of delta-matroids that have a twist of width at most one.

To state the Rough Structure Theorem we need the following. Let $D=(E,\mathcal{F})$ be a delta-matroid and let $\mathcal{F}_{\min}$ be the set of feasible sets of minimum size. Then $D_{\min}:=(E,\mathcal{F}_{\min})$ is a matroid. For a matroid $M$ with ground set $E$, a subset $A$ of $E$ is said to be a separator of $M$ if $A$ is a union of components of $M$. Note that both $\emptyset$ and $E$ are always separators. In terms of the matroid rank function, where the rank $r(X)$ of a set $X\subseteq E$ is defined to be the size of the largest intersection of $X$ with a basis of $M$, the set $A$ is a separator if and only if $r(A)+r(E-A)=r(M)$. Throughout the paper we use $\overline{A}$ for the complement $E-A$ of $A$, and $D|X$ denotes the restriction of $D$ to $X\subseteq E$ (see the beginning of Section 2 for its definition).

We now state the first of our two main results: a Rough Structure Theorem for delta-matroids admitting a twist of width one.

Theorem 1.1**.**

Let $D=(E,\mathcal{F})$ be a delta-matroid. Then $D$ has a twist of width one if and only if there is some $A\subseteq E$ such that

1. (1)

$A$* is a separator of $D_{\min}$,* 2. (2)

$D|A$* is a matroid, and* 3. (3)

$D|\overline{A}$* is of width one.*

We actually prove a result that is stronger than Theorem 1.1. This stronger result appears below as Theorem 2.3 and the present theorem follows immediately from it.

As an application of Theorem 1.1, we find an excluded minor characterisation of the class of delta-matroids that have a twist of width one as our second main result, Theorem 1.3. This class of delta-matroids is shown to be minor closed in Proposition 3.1, and its set of excluded minors comprises the delta-matroids in the following definition together with their twists.

Definition 1.2**.**

Let $D_{1}$ denote the delta-matroid on the elements $a,b$ with feasible sets

[TABLE]

For $i=2,\ldots,5$ let $D_{i}$ denote the delta-matroid on the elements $a,b,c$ with feasible sets given by

[TABLE]

Throughout this paper $D_{1},\ldots,D_{5}$ refer exclusively to these delta-matroids. Let $\mathcal{D}_{[5]}$ be the set of all twists of these delta-matroids. Note that $D_{i}\in\mathcal{D}_{[5]}$ for all $i\in\{1,2,\dots,5\}$ via the empty twist.

Theorem 1.3**.**

A delta-matroid has a twist of width at most one if and only if it has no minor isomorphic to a member of $\mathcal{D}_{[5]}$.

The proof of this theorem appears at the end of Section 3.

We note that the excluded minors of twists of matroids (i.e., twists of width zero delta-matroids) has been shown, but not explicitly stated, to be $(\{a\},\{\emptyset,\{a\}\})$, $D_{3}$, and $D_{3}\ast\{a\}$ by A. Duchamp in [6]. This result can be recovered from Theorem 1.3 by restricting to even delta-matroids, where an even delta-matroid is a delta-matroid in which the difference in size between any two feasible sets is even.

Above we mentioned the close connection between delta-matroids and graphs in surfaces. The width of a delta-matroid can be viewed as the analogue of the genus (or more precisely the Euler genus) of an embedded graph, while twisting is the analogue of S. Chmutov’s partial duality of [3]. Thus characterising twists of width one is the analogue of characterising partial duals of graphs in the real projective plane. The topological graph theoretical analogues of Theorems 1.1 and 1.3 can be found in [7, 8].

2. The proof of the Rough Structure Theorem

For the convenience of the reader, we recall some standard matroid and delta-matroid terminology. Given a delta-matroid $D=(E,\mathcal{F})$ and element $e\in E$, if $e$ is in every feasible set of $D$ then we say that $e$ is a coloop of $D$. If $e$ is in no feasible set of $D$, then we say that $e$ is a loop of $D$. If $e\in E$ is not a coloop, then $D$ delete $e$, denoted by $D\setminus e$, is the delta-matroid $(E-e,\{F:F\in\mathcal{F}\text{ and }F\subseteq E-e\})$. If $e\in E$ is not a loop, then $D$ contract $e$, denoted by $D/e$, is the delta-matroid $(E-e,\{F-e:F\in\mathcal{F}\text{ and }e\in F\})$. If $e\in E$ is a loop or coloop, then $D/e=D\setminus e$. Useful identities that we use frequently are $D/e=(D\ast e)\setminus e$ and $D\setminus e=(D\ast e)/e$. If $D^{\prime}$ is a delta-matroid obtained from $D$ by a sequence of deletions and contractions, then $D^{\prime}$ is independent of the order of the deletions and contractions used in its construction, so we can define $D\setminus X/Y$ for disjoint subsets $X$ and $Y$ of $E$, as the result of deleting each element in $X$ and contracting each element in $Y$ in some order. A minor of $D$ is any delta-matroid that is obtained from it by deleting or contracting some of its elements. The restriction of $D$ to a subset $A$ of $E$, written $D|A$, is equal to $D\setminus\overline{A}$. Note that if $\emptyset\in\mathcal{F}(D)$ then $F$ is feasible in $D|A$ if and only if $F\subseteq A$ and $F\in\mathcal{F}(D)$.

The connectivity function $\lambda_{M}$ of a matroid $M$ on ground set $E$ with rank function $r$ is defined on all subsets $A$ of $E$ by $\lambda_{M}(A)=r(A)+r(\overline{A})-r(E)$. Recall that $A$ is said to be a separator of $M$ if $A$ is a union of components of $M$. This happens if and only if $\lambda_{M}(A)=0$. Moreover, $A$ is a separator if and only if $\overline{A}$ is a separator.

We will use Bouchet’s analogue of the rank function for delta-matroids from [2]. For a delta-matroid $D=(E,\mathcal{F})$, it is denoted by $\rho_{D}$ or simply $\rho$ when $D$ is clear from the context. Its value on a subset $A$ of $E$ is given by

[TABLE]

The following theorem determines the width of a twist of a delta-matroid.

Theorem 2.1**.**

Let $D=(E,\mathcal{F})$ be a delta-matroid and $A\subseteq E$. Then the width, $w(D*A)$, of the twist of $D$ by $A$ is given by

[TABLE]

Proof.

The largest feasible set in $D*A$ has size $\max\{|F\bigtriangleup A|:F\in\mathcal{F}(D)\}$. Take $F^{\prime}\in\mathcal{F}$ such that $|F^{\prime}\bigtriangleup A|$ is maximal. Then $|F^{\prime}\bigtriangleup\overline{A}|$ is minimal. As $\rho(\overline{A})=|E|-\min\{|F\bigtriangleup\overline{A}|:F\in\mathcal{F}\}$, we see that $\rho(\overline{A})=|E|-|F^{\prime}\bigtriangleup\overline{A}|=|F^{\prime}\bigtriangleup A|$. Hence the largest feasible set in $D*A$ has size equal to $\rho(\overline{A})$.

Next, the size of the smallest feasible set in $D*A$ is $|E|$ minus the size of the largest feasible set in $(D*A)^{*}=D*\overline{A}$. By an application of the above, it follows that the size of the smallest feasible set in $D*A$ is $|E|-\rho(A)$. Hence $w(D*A)=\rho(\overline{A})-|E|+\rho(A)$.

We let $r$ and $n$ be the rank and nullity functions, respectively, of $D_{\min}$. From [4], we know that $w(D|A)=\rho(A)-r(A)-n(E)+n(A)$. As $n(A)=|A|-r(A)$ and $n(E)=|E|-r(E)$,

[TABLE]

giving the result. ∎

The following two theorems are immediate consequences of Theorem 2.1. The Rough Structure Theorem, Theorem 1.1, follows immediately from the second of them.

Theorem 2.2** (Chun et al [5]).**

Let $D=(E,\mathcal{F})$ be a delta-matroid, $A\subseteq E$, and $\overline{A}=E-A$. Then $D*A$ is a matroid if and only if $A$ is a separator of $D_{\min}$, and both $D|A$ and $D|\overline{A}$ are matroids.

Theorem 2.3**.**

Let $D=(E,\mathcal{F})$ be a delta-matroid, $A\subseteq E$, and $\overline{A}=E-A$. Then $D*A$ has width one if and only if $A$ is a separator of $D_{\min}$, and one of $D|A$ and $D|\overline{A}$ is a matroid and the other has width one.

For convenience, we write down the following straightforward corollary. It provides the form of the Rough Structure Theorem that we use to find excluded minors in the next section.

Corollary 2.4**.**

Let $D=(E,\mathcal{F})$ be a delta-matroid in which $\emptyset$ is feasible. Then the following hold.

1. (1)

$D$* has a twist of width zero if and only if there exists $A\subseteq E$ such that $D|A$ and $D|\overline{A}$ are both of width zero.* 2. (2)

$D$* has a twist of width one if and only if there exists $A\subseteq E$ such that $D|A$ is a matroid, and $D|\overline{A}$ is of width one.*

Proof.

This is a straightforward consequence of the fact that if $\emptyset$ is feasible in $D$, then $D_{\min}$ is the matroid on $E(D)$ where each element is a loop, thus every set $A\subseteq E$ is a separator of $D_{\min}$. ∎

3. The proof of the excluded minor characterisation

We begin this section by verifying that the class of delta-matroids in question is indeed minor-closed.

Proposition 3.1**.**

For each $k\in\mathbb{N}_{0}$, the set of delta-matroids with a twist of width at most $k$ is minor-closed.

Proof.

Let $D=(E,\mathcal{F})$ and suppose $w(D\ast A)\leq k$ for some $A\subseteq E$. If $E$ is empty the result is trivial, so assume not and let $e\in E$. If $e\notin A$ then $(D\setminus e)\ast A=(D\ast A)\setminus e$, and $(D/e)*A=((D*e)\setminus e)*A=((D*e)*A)\setminus e=((D*A)\ast e)\setminus e=(D*A)/e$. Similarly, if $e\in A$ then $e\notin A-e$, so using and extending the previous argument, $(D/e)\ast(A-e)=(D*(A-e))/e=((D*A)*e)/e=(D*A)\setminus e$, and $(D\setminus e)\ast(A-e)=(D\ast(A-e))\setminus e=((D\ast A)\ast e)\setminus e=(D\ast A)/e$. In each case we see that $D/e$ and $D\setminus e$ have a twist that can be written as $(D\ast A)/e$ or $(D\ast A)\setminus e$. Since deletion and contraction never increase width it follows that $D/e$ and $D\setminus e$ have twists of width at most $w(D\ast A)\leq k$. The result follows. ∎

Lemma 3.2**.**

Let $D=(E,\mathcal{F})$ be a delta-matroid and $A\subseteq E$. Then

[TABLE]

Proof.

In the proof of Proposition 3.1 it was shown that if $e\notin A$ then $(D*A)/e=(D/e)*A$ and $(D\ast A)\setminus e=(D\setminus e)\ast A$, whereas if $e\in A$ then $(D*A)\setminus e=(D/e)\ast(A-e)$ and $(D\ast A)/e=(D\setminus e)\ast(A-e)$. The result follows immediately from this. ∎

Lemma 3.3**.**

Let $D$ be a delta-matroid in which the empty set is feasible. Then $D$ has a twist of width at most 1, or contains a minor isomorphic to one of $D_{1},\ldots,D_{5}$.

Proof.

For any delta-matroid $D$ in which the empty set is feasible, set

[TABLE]

(Technically we should record the fact that $L$ depends upon $D$ in the notation, however we avoid doing this for notational simplicity. This should cause no confusion.) Note that $L$ may be empty. Construct a (simple) graph $G_{D}$ as follows. Take one vertex $v_{x}$ for each element $x\in\overline{L}$, and add one other vertex $v_{L}$. The edges of $G_{D}$ arise from certain two-element feasible sets of $D$. Add an edge $v_{x}v_{y}$ to $G_{D}$ for each pair $x,y\in\overline{L}$ with $\{x,y\}\in\mathcal{F}(D)$; add an edge $v_{x}v_{L}$ to $G_{D}$ if $\{x,z\}\in\mathcal{F}(D)$ for some $z\in L$.

We consider two cases: when $G_{D}$ is bipartite, and when it is not. We will show that if $G_{D}$ is bipartite then $D$ must have a twist of width at most one or a minor isomorphic to $D_{1}$ or $D_{2}$; if $G_{D}$ is not bipartite then it must have a minor isomorphic to $D_{1}$, $D_{3}$, $D_{4}$, or $D_{5}$.

Case 1. Let $D$ be a delta-matroid in which the empty set is feasible, and such that $G_{D}$ is bipartite. Fix a 2-colouring of $G_{D}$. Let $A$ be the set of elements in $E(D)$ that correspond to the vertices in the colour class containing $v_{L}$ together with the elements in $L$, and let $\overline{A}\subseteq E(D)$ be the set of elements corresponding to the vertices in the colour class not containing $v_{L}$.

We start by showing

[TABLE]

where $U_{0,|\overline{A}|}$ denotes the uniform matroid with rank zero and $|\overline{A}|$ elements.

To see why (1) holds, note that $\mathcal{F}(D|\overline{A})=\{F:F\subseteq\overline{A}\text{ and }F\in\mathcal{F}(D)\}$. Since the elements in $\overline{A}$ correspond to vertices in $\overline{L}$, no feasible sets of $D|\overline{A}$ have size one. Furthermore, $\mathcal{F}(D|\overline{A})$ cannot contain any sets of size two since, by the construction of $G_{D}$, whenever $\{x,y\}\in\mathcal{F}(D)$ the corresponding vertices $v_{x}$ and $v_{y}$ are in different colour classes. Since $\emptyset\in\mathcal{F}(D|\overline{A})$, the Symmetric Exchange Axiom ensures that there are no other feasible sets. (If $F\in\mathcal{F}(D|\overline{A})$ with $F\neq\emptyset$, take $x\in\emptyset\bigtriangleup F$. Then by the Symmetric Exchange Axiom $\emptyset\bigtriangleup\{x,y\}$ must be in $\mathcal{F}(D|\overline{A})$ for some $y$, but there are no feasible sets of size one or two.) This completes the justification of (1).

Next we examine the feasible sets in $D|A$. Trivially $\emptyset\in\mathcal{F}(D|A)$. The set of feasible sets of $D|A$ of size one is $\{F\in\mathcal{F}(D|A):|F|=1\}=\{F\in\mathcal{F}(D):|F|=1\}=\{\{x\}:x\in L\}$.

If $\mathcal{F}(D|A)$ contains a set $\{x,y\}$ of size two then $x,y\in L$ as otherwise there would be an edge $v_{x}v_{y}$ in $G_{D}$ whose ends are in the same colour class. It follows in this case that $D|A$ and hence $D$ contains a minor isomorphic to $D_{1}$.

Now assume that $\mathcal{F}(D|A)$ does not contain a set of size two. If $\mathcal{F}(D|A)$ has no sets of size one then, arguing via the Symmetric Exchange Axiom as in the justification of (1), we have $D|A\cong U_{0,|A|}$. Taken together with (1), this implies that $A$ satisfies the conditions of the first part of Corollary 2.4, so $D$ has a twist of width zero.

Suppose that $\mathcal{F}(D|A)$ does contain a set of size one. If it contains no sets of size greater than one then $D|A$ is of width one, and by combining this with (1), it follows from Corollary 2.4 that $D$ has a twist of width one ($D\ast A$ and $D\ast\overline{A}$ are such twists). On the other hand, if $\mathcal{F}(D|A)$ does contain a set of size greater than one, then, as it does not contain a set of size two, the Symmetric Exchange Axiom guarantees there is a set in $\mathcal{F}(D|A)$ of size exactly three. (If not, let $F$ be a minimum sized feasible set with $|F|>3$. Then $F\setminus\{x,y\}$ is feasible and of size at least two for some $x,y\in\emptyset\bigtriangleup F$ contradicting the minimality of $|F|>3$.) Let $\{x,y,z\}\in\mathcal{F}(D|A)$. Then after possibly relabelling its elements, the collection of feasible sets of $D|\{x,y,z\}$ is one of

[TABLE]

Only the first of the three cases is possible as the Symmetric Exchange Axiom fails for the other two showing that neither is the collection of feasible sets of a delta-matroid. Hence, restricting $D$ to $\{x,y,z\}$ results in a minor isomorphic to $D_{2}$.

Thus we have shown that if $G_{D}$ is bipartite then $D$ has a twist of width at most one or contains a minor isomorphic to $D_{1}$ or $D_{2}$. This completes the proof of Case 1.

Case 2. Let $D$ be a delta-matroid in which the empty set is feasible, and such that $G_{D}$ is non-bipartite. We will show that $D$ contains a minor isomorphic to one of $D_{1}$, $D_{3}$, $D_{4}$ or $D_{5}$ by induction on the length of a shortest odd cycle in $G_{D}$.

For the base of the induction suppose that $G_{D}$ has an odd cycle $C$ of length three. There are two sub-cases, when $v_{L}$ is not in $C$ and when it is. Note that the former sub-case includes the situation where $L=\emptyset$.

Sub-case 2.1. Suppose that $v_{L}$ is not in $C$. Let $x,y,z\in E(D)$ be the elements corresponding to the three vertices of $C$. We have $x,y,z\in\overline{L}$, so $\{x\},\{y\},\{z\}\notin\mathcal{F}(D)$. From the three edges of $C$ we have $\{x,y\},\{y,z\},\{z,x\}\in\mathcal{F}(D)$. It follows that $D|\{x,y,z\}$ is isomorphic to either $D_{3}$ or $D_{4}$ giving the required minor.

Sub-case 2.2. Suppose that $v_{L}$ is in $C$. Let $v_{x},v_{y},v_{L}$ be the vertices in $C$. The edges of $C$ give that $\{x,y\}\in\mathcal{F}(D)$, and since $x,y\in\overline{L}$ we have $\{x\},\{y\}\notin\mathcal{F}(D)$. We also know that there are elements $\alpha,\beta\in L$ such that $\{\alpha\},\{\beta\},\{x,\alpha\},\{y,\beta\}\in\mathcal{F}(D)$, where possibly $\alpha=\beta$.

If $\alpha=\beta$ then $D|\{x,y,\alpha\}$ must have feasible sets

[TABLE]

The first case gives a minor of $D$ isomorphic to $D_{5}$; in the second case, $(D|\{x,y,\alpha\})/\alpha$ is a minor of $D$ isomorphic to $D_{1}$.

If $\alpha\neq\beta$ then the feasible sets of $D|\{x,y,\alpha,\beta\}$ of size zero or one are exactly $\emptyset$, $\{\alpha\}$, and $\{\beta\}$. From $G_{D}$, the feasible sets of size two include $\{x,\alpha\},\{y,\beta\},\{x,y\}$. If $\{y,\alpha\}$ is also feasible then $D|\{x,y,\alpha\}$ is isomorphic to one of the delta-matroids arising from (2), so $D$ has a minor isomorphic to $D_{1}$ or $D_{5}$. The case when $\{x,\beta\}$ is feasible is similar. If $\{\alpha,\beta\}$ is feasible then $D|\{\alpha,\beta\}$ is isomorphic to $D_{1}$.

The case that remains is when the feasible sets of $D|\{x,y,\alpha,\beta\}$ of size at most two are exactly

[TABLE]

By applying the Symmetric Exchange Axiom to each of the pairs of feasible sets $(\{\alpha\},\{x,y\})$, $(\{\beta\},\{x,y\})$, $(\{\beta\},\{x,\alpha\})$ and $(\{\alpha\},\{y,\beta\})$, one can show that each of the three element sets,

[TABLE]

is feasible in $D|\{x,y,\alpha,\beta\}$. Finally, $\{\alpha,\beta,x,y\}$ may or may not be feasible.

If $\{\alpha,\beta,x,y\}$ is feasible then $(D|\{x,y,\alpha,\beta\})/\{x,y\}$ is isomorphic to $D_{1}$; if $\{\alpha,\beta,x,y\}$ is not feasible then $(D|\{x,y,\alpha,\beta\})/\{\alpha\}$ is isomorphic to $D_{5}$.

This completes the base of the induction.

For the inductive hypothesis, we assume that, for some $n>3$, if $D$ is a delta-matroid such that $\emptyset\in\mathcal{F}(D)$ and $G_{D}$ has an odd cycle of length less than $n$, then $D$ has a minor isomorphic to $D_{1},D_{3},D_{4}$, or $D_{5}$.

Suppose that $\emptyset\in\mathcal{F}(D)$ and a shortest odd cycle $C$ of $G_{D}$ has length $n$. Again there are two sub-cases: when $v_{L}$ is not in $C$ and when it is.

Sub-case 2.3. Suppose that $v_{L}$ is not in $C$. Let $C=v_{x_{1}}v_{x_{2}}\ldots v_{x_{n}}v_{x_{1}}$. Since each $x_{i}\in\overline{L}$ and $C$ is the shortest odd cycle in $G_{D}$,

[TABLE]

is a complete list of the feasible sets of size at most two in $D|\{x_{1},\ldots,x_{n}\}$.

Next, we show

[TABLE]

To see why (4) holds, first note that, since $n>3$, every set of three distinct vertices in the cycle includes a non-adjacent pair. If $\{x_{i},x_{j},x_{k}\}$ were feasible in $D|\{x_{1},\ldots,x_{n}\}$, then, without loss of generality, $\{x_{j},x_{k}\}\notin\mathcal{F}(D|\{x_{1},\ldots,x_{n}\})$. As $x_{i}\in\{x_{i},x_{j},x_{k}\}\bigtriangleup\emptyset$, an application of the Symmetric Exchange Axiom would imply that $\{x_{i},x_{j},x_{k}\}\bigtriangleup\{x_{i},z\}$ is feasible for some $z\in\{x_{i},x_{j},x_{k}\}$. Thus $\{x_{j},x_{k}\},\{x_{j}\}$, or $\{x_{k}\}$ would be feasible, a contradiction to (3). Thus (4) holds.

Next we show that, taking indices modulo $n$,

[TABLE]

for any $i$ and $j$ such that $1\leq i,j\leq n$ and $i,i+1,j,j+1$ are pairwise distinct.

For this, first suppose that neither $x_{i+1}$ and $x_{j}$ nor $x_{j+1}$ and $x_{i}$ are adjacent in $C$. Then by (3), $\{x_{i},x_{i+1}\}$ and $\{x_{j},x_{j+1}\}$ are feasible. As $x_{j}$ is in their symmetric difference, by the Symmetric Exchange Axiom, $\{x_{i},x_{i+1}\}\bigtriangleup\{x_{j},y\}$ is feasible for some $y\in\{x_{i},x_{i+1},x_{j},x_{j+1}\}$. Thus $\{x_{i},x_{i+1},x_{j}\},\{x_{i},x_{j}\},\{x_{i+1},x_{j}\}$ or $\{x_{i},x_{i+1},x_{j},x_{j+1}\}$ is feasible. By (3) and (4), $\{x_{i},x_{i+1},x_{j},x_{j+1}\}$ is feasible. If $x_{i+1}$ and $x_{j}$ are adjacent then the Symmetric Exchange Axiom implies that $\{x_{i},x_{i+1}\}\bigtriangleup\{x_{i+3},z\}$ is feasible for some $z\in\{x_{i},x_{i+1},x_{i+2},x_{i+3}\}$. Again, (3) and (4) imply that $\{x_{i},x_{i+1},x_{i+2},x_{i+3}\}$ must be feasible. The other case is identical. This completes the justification of (5).

Combining (3)–(5) gives that all of $\emptyset,\{x_{1},x_{2}\},\{x_{2},x_{3}\},\ldots,\{x_{n-2},x_{1}\}$, but none of $\{x_{1}\},\ldots,\{x_{n-2}\}$, are feasible in $(D|\{x_{1},\ldots,x_{n}\})/\{x_{n-1},x_{n}\}$. Hence the graph $G_{(D|\{x_{1},\ldots,x_{n}\})/\{x_{n-1},x_{n}\}}$ has a shorter odd cycle than $G_{D}$. By the inductive hypothesis, $(D|\{x_{1},\ldots,x_{n}\})/\{x_{n-1},x_{n}\}$ and hence $D$ has a minor isomorphic to one of $D_{1}$, $D_{3}$, $D_{4}$ or $D_{5}$.

Sub-case 2.4. Suppose that $v_{L}$ is in $C$. Let $C=v_{L}v_{x_{2}}v_{x_{3}}\dots v_{x_{n}}v_{L}$. The edges of the cycle give that, for each $2\leq i\leq n-1$, $\{x_{i},x_{i+1}\}\in\mathcal{F}(D)$. Also, for $2\leq i\leq n$, since $x_{i}\in\overline{L}$ we have $\{x_{i}\}\notin\mathcal{F}(D)$. We also know that there are elements $\alpha,\beta\in L$ such that $\{\alpha\},\{\beta\},\{\alpha,x_{2}\},\{\beta,x_{n}\}\in\mathcal{F}(D)$ where possibly $\alpha=\beta$. (This possibility is covered in the following analysis.)

When $\alpha\neq\beta$, if $\{\alpha,\beta\}\in\mathcal{F}(D)$, then $D|\{\alpha,\beta\}$ is isomorphic to $D_{1}$, therefore we assume $\{\alpha,\beta\}\notin\mathcal{F}(D)$. Using that $C$ is a shortest odd cycle, the feasible sets of $D|\{\alpha,\beta,x_{2},\ldots,x_{n}\}$ of size at most two are exactly

[TABLE]

An argument similar to the justification of (4) gives that

[TABLE]

However

[TABLE]

To see this note that $x_{n-1}\in\{\alpha\}\bigtriangleup\{x_{n-1},x_{n}\}$, so the Symmetric Exchange Axiom gives that one of $\{\alpha,x_{n-1}\}$, $\{x_{n-1}\}$, or $\{\alpha,x_{n-1},x_{n}\}$ is feasible, and we know from (6) that the feasible set must be the third option. That $\{\beta,x_{n-1},x_{n}\}$ is feasible follows from a similar argument.

We next show that for each $2\leq i<n-2$,

[TABLE]

For this, first consider $x_{2}\in\{x_{n-1},x_{n}\}\bigtriangleup\{\alpha,x_{2}\}$. The Symmetric Exchange Axiom implies that $\{x_{n-1},x_{n}\}\bigtriangleup\{x_{2},z\}$ is feasible for some $z\in\{\alpha,x_{2},x_{n-1},x_{n}\}$. By (6) and (7), $z=\alpha$, thus $\{\alpha,x_{2},x_{n-1},x_{n}\}$ is feasible. Next, to show that $\{x_{i},x_{i+1},x_{n-1},x_{n}\}$ is feasible, we take $x_{i}\in\{x_{n-1},x_{n}\}\bigtriangleup\{x_{i},x_{i+1}\}$ and apply the Symmetric Exchange Axiom as above to see that $\{x_{n-1},x_{n}\}\bigtriangleup\{x_{i},z\}$ is feasible, where $z$ must equal $x_{i+1}$. Lastly, to show that $\{\beta,x_{n-2},x_{n-1},x_{n}\}$ is feasible, we first show that $\{\beta,x_{n-2},x_{n}\}\notin\mathcal{F}(D|\{\alpha,\beta,x_{2},\ldots,x_{n}\})$. If $\{\beta,x_{n-2},x_{n}\}$ were feasible, then since $x_{n-2}\in\emptyset\bigtriangleup\{\beta,x_{n-2},x_{n}\}$, the Symmetric Exchange Axiom would give $\{x_{n-2}\}$, $\{\beta,x_{n-2}\}$ or $\{x_{n-2},x_{n}\}$ as feasible, a contradiction. Now showing that $\{\beta,x_{n-2},x_{n-1},x_{n}\}$ is feasible comes from taking $x_{n-2}\in\{\beta,x_{n}\}\bigtriangleup\{x_{n-2},x_{n-1}\}$. The Symmetric Exchange Axiom gives that $\{\beta,x_{n}\}\bigtriangleup\{x_{n-2},z\}$ is feasible for some $z\in\{\beta,x_{n-2},x_{n-1},x_{n}\}$, of which $z=x_{n-1}$ is the only possibility.

From (6)–(9) it follows that all of $\emptyset$ , $\{\alpha\}$, $\{\beta\}$, $\{\alpha,x_{2}\}$, $\{x_{2},x_{3}\}$, $\ldots$, $\{\beta,x_{n-2}\}$, but none of $\{x_{2}\}$, $\ldots$, $\{x_{n-2}\}$, are feasible in $(D|\{\alpha,x_{2},\ldots,x_{n},\beta\})/\{x_{n-1},x_{n}\}$. Hence the graph $G_{(D|\{\alpha,x_{2},\ldots,x_{n},\beta\})/\{x_{n-1},x_{n}\}}$ has a shorter odd cycle than $G_{D}$. The inductive hypothesis gives that $(D|\{\alpha,x_{2},\ldots,x_{n},\beta\})/\{x_{n-1},x_{n}\}$ and hence $D$ has a minor isomorphic to one of $D_{1}$, $D_{3}$, $D_{4}$ or $D_{5}$. This completes the proof of the sub-case, and the lemma. ∎

We now apply Lemma 3.3 to prove our excluded minor characterisation of the family of delta-matroids admitting a twist of width at most one.

Proof of Theorem 1.3.

All twists of the delta-matroids $D_{1},\ldots,D_{5}$ are of width at least two. Since the set of delta-matroids with a twist of width at most one is minor-closed it follows that no minor of a delta-matroid with a twist of width at most one is isomorphic to a member of $\mathcal{D}_{[5]}$. This proves one direction of the theorem.

Conversely suppose that every twist of a delta-matroid $D=(E,\mathcal{F})$ is of width at least two. Let $A\in\mathcal{F}$. Then $D*A$ is a delta-matroid in which $\emptyset$ is feasible and in which every twist is of width at least two. By Lemma 3.3, $D*A$ has a minor isomorphic to one of $D_{1},\ldots,D_{5}$. It follows from Lemma 3.2 that $D$ has a minor isomorphic to a member of $\mathcal{D}_{[5]}$. ∎