Minimal Braces

TL;DR

This paper introduces a new induction tool for minimal braces, establishing an upper size bound for minimal braces of order 2n and characterizing those meeting this bound.

Contribution

It derives a main theorem for minimal braces based on McCuaig's brace generation theorem and applies it to characterize extremal minimal braces.

Findings

Minimal braces of order 2n have size at most 5n-10 for n ≥ 6.

Complete characterization of minimal braces that meet the size bound.

Main theorem serves as an induction tool for minimal braces.

Abstract

McCuaig (2001, Brace Generation, J. Graph Theory 38: 124-169) proved a generation theorem for braces, and used it as the principal induction tool to obtain a structural characterization of Pfaffian braces (2004, P{\'o}lya's Permanent Problem, Electronic J. Combinatorics 11: R79). A brace is minimal if deleting any edge results in a graph that is not a brace. From McCuaig's brace generation theorem, we derive our main theorem that may be viewed as an induction tool for minimal braces. As an application, we prove that a minimal brace of order $2n$ has size at most $5n-10$, when $n \geq 6$, and we provide a complete characterization of minimal braces that meet this upper bound. A similar work has already been done in the context of minimal bricks by Norine and Thomas (2006, Minimal Bricks, J. Combin. Theory Ser. B 96: 505-513) wherein they deduce the main result from the brick generation theorem due to the same authors (2007, Generating Bricks, J. Combin. Theory Ser. B 97: 769-817).

Full text

A brace is minimal if deleting any edge results in a graph that is not a brace. From McCuaig’s brace generation theorem, we derive our main theorem that may be viewed as an induction tool for minimal braces. As an application, we prove that a minimal brace of order $2n$ has size at most $5n-10$, when $n\geq 6$, and we provide a complete characterization of minimal braces that meet this upper bound.

A similar work has already been done in the context of minimal bricks by Norine and Thomas (2006, Minimal Bricks, J. Combin. Theory Ser. B 96: 505-513) wherein they deduce the main result from the brick generation theorem due to the same authors (2007, Generating Bricks, J. Combin. Theory Ser. B 97: 769-817).

In memory of Robin Thomas

1 Bipartite matching covered graphs

For general graph-theoretic notation and terminology, we refer the reader to Bondy and Murty [1]. All graphs considered in this paper are finite and loopless; however, we do allow multiple (i.e., parallel) edges. For a graph $G$, its order is the number of vertices (i.e., $|V(G)|$), and its size is the number of edges (i.e., $|E(G)|$). For a subset $X$ of $V(G)$, we denote by $\partial(X)$ the cut associated with $X$, and we refer to $X$ and $\overline{X}:=V(G)-X$ as the shores of $\partial(X)$. Thus $\partial(X)$ is the set of edges that have exactly one end in either shore. A cut is trivial if either of its shores is a singleton. The graph obtained by contracting the shore $X$ to a single vertex $x$ is denoted by $G/(X\rightarrow x)$, or simply by $G/X$. The two graphs $G/X$ and $G/\overline{X}$ are called the $\partial(X)$-contractions of $G$.

A connected graph $G$ is $k$-extendable if it has a matching of cardinality $k$, and if each such matching extends to (i.e., is a subset of) a perfect matching of $G$. For a comprehensive treatment of matching theory and its origins, we refer the reader to Lovász and Plummer [6]. All graphs considered in this paper are $1$-extendable, and we shall instead refer to them as matching covered graphs. It is easily seen that these graphs (of order four or more) are $2$-connected. Also, for a graph $G$, we let $n_{G}:=\frac{|V(G)|}{2}$ and $m_{G}:=|E(G)|$; whence $G$ has order $2\cdot n_{G}$ and size $m_{G}$.

A cut $\partial(X)$ of $G$ is tight if $|M\cap\partial(X)|=1$ for each perfect matching $M$ of $G$. A matching covered graph that is free of nontrivial tight cuts is called a brace if it is bipartite, or otherwise a brick. It is easily verified that if $\partial(X)$ is a nontrivial tight cut of a matching covered graph $G$ then each $\partial(X)$-contraction of $G$ is a matching covered graph of strictly smaller order. This observation leads to a decomposition of any matching covered graph into a list of bricks and braces; this procedure is known as a tight cut decomposition of $G$. Clearly, a graph may admit several tight cut decompositions. However, Lovász [5] proved the remarkable result that any two tight cut decompositions of a matching covered graph $G$ yield the same list of bricks and braces (except possibly for the multiplicities of edges). We remark that $G$ is bipartite if and only if its tight cut decomposition yields only braces (i.e., it yields no bricks). The bipartite matching covered graph $Q_{10}$, shown in Figure 1(a), has a nontrivial tight cut $\partial(X)$, and each of its $\partial(X)$-contractions is isomorphic to the brace $K_{3,3}$.

Several important properties of a matching covered graph $G$ may be deduced by analysing its bricks and braces. (For instance, $G$ is Pfaffian if and only if each of its bricks and braces is Pfaffian; see [13, 4].) Consequently, researchers were led to gain a deeper understanding of bricks and braces. McCuaig [7] established a generation theorem for simple braces, and used this as the principal induction tool to obtain a structural characterization of Pfaffian braces [8]. Robertson, Seymour and Thomas [12] arrived at the same characterization using a different approach. (These groundbreaking works led to a polynomial-time algorithm for deciding whether or not a given bipartite graph is Pfaffian; see [9].)

A brace is minimal if deleting any edge results in a graph that is not a brace. The aforementioned McCuaig’s Theorem is a powerful induction tool for the class of simple braces. The object of this paper is to use McCuaig’s Theorem to derive an induction tool for the class of minimal braces — a proper subset of the class of simple braces. In the following three subsections, we introduce the necessary terminology to make this more precise.

A similar work has already been done in the context of minimal bricks by Norine and Thomas [10], wherein they deduce the main result from the brick generation theorem due to the same authors [11].

1.1 Braces

For a connected bipartite graph $G$, we adopt the notation $G[A,B]$ to denote its color classes. We will generally use letters $A$ and $B$ to denote the color classes; sometimes we may instead use $A^{\prime}$ and $B^{\prime}$. As shown in Figure 3, members of $A$ (or of $A^{\prime}$) will be denoted using letters $a$ and $w$ (with subscripts and/or superscripts) and will be depicted using hollow nodes; likewise, members of $B$ (or of $B^{\prime}$) will be denoted using letters $b$ and $u$ (with subscripts and/or superscripts) and will be depicted using solid nodes.

The neighborhood of a set of vertices $Z$ is denoted by $N_{G}(Z)$. The following may be deduced from the well-known Hall’s Theorem.

Proposition 1.1

For a connected bipartite graph $G[A,B]$, where $|A|=|B|$, the following are equivalent:

1. (i)

$G$* is matching covered,* 2. (ii)

$|N_{G}(Z)|\geq|Z|+1$* for every nonempty proper subset $Z$ of $A$, and* 3. (iii)

$G-a-b$* has a perfect matching for each pair of vertices $a\in A$ and $b\in B$. *$\Box$

Suppose that $X$ is an odd subset of the vertex set of a connected bipartite graph $G[A,B]$. Then one of the two sets $A\cap X$ and $B\cap X$ is larger than the other; the larger set, denoted $X_{+}$, is called the majority part of $X$; the smaller set, denoted $X_{-}$, is called the minority part of $X$. The following proposition provides a convenient way of visualizing tight cuts in bipartite graphs. It is easily proved. (See Figure 1(a) for an example.)

Proposition 1.2

A cut $\partial(X)$ of a bipartite matching covered graph $G[A,B]$ is tight if and only if the following hold:

1. (i)

$|X|$* is odd and $|X_{+}|=|X_{-}|+1$; consequently $|\overline{X}_{+}|=|\overline{X}_{-}|+1$; and* 2. (ii)

*there are no edges between $X_{-}$ and $\overline{X}_{-}$. *$\Box$

Recall that a brace is a bipartite matching covered graph that is free of nontrivial tight cuts. The following characterization of braces may be deduced from Proposition 1.2.

Proposition 1.3

For a connected bipartite graph $G[A,B]$ of order six or more, where $|A|=|B|$, the following are equivalent:

1. (i)

$G$* is a brace,* 2. (ii)

$|N_{G}(Z)|\geq|Z|+2$* for every nonempty subset $Z$ of $A$ such that $|Z|<|A|-1$,* 3. (iii)

$G-a_{1}-a_{2}-b_{1}-b_{2}$* has a perfect matching for any four distinct vertices $a_{1},a_{2}\in A$ and $b_{1},b_{2}\in B$,* 4. (iv)

$G$* is $2$-extendable. *$\Box$

Thus braces (of order at least four) are precisely those bipartite graphs that are $2$-extendable. The following immediate consequence of Propositions 1.1 and 1.3 is worth noting.

Corollary 1.4

Let $G$ denote a bipartite graph obtained from a graph $H$ by adding an edge. If $H$ is matching covered then so is $G$. Furthermore, if $H$ is a brace then so is $G$. $\Box$

The braces $K_{2}$ and $C_{4}$ are the only simple bipartite matching covered graphs of order at most four. For a bipartite matching covered graph $G$ of order six or more, it is easy to show that if $G$ has a $2$-vertex-cut then $G$ has a nontrivial tight cut. This implies the following.

Proposition 1.5

Every brace, of order six or more, is $3$-connected. $\Box$

A vertex is cubic if its degree equals three; it is noncubic if it has degree four or more. A graph is cubic if each of its vertices is cubic, and it is noncubic if it has a noncubic vertex.

McCuaig [7] described three infinite families of simple braces: prisms222McCuaig [7] refers to ‘prisms’ as ‘ladders’., Möbius ladders and biwheels. A biwheel of order $2n$ (where $n\geq 4$), denoted $B_{2n}$, is the simple bipartite graph obtained from the cycle graph $C_{2n-2}$ by adding two nonadjacent vertices — each of which has degree exactly $n-1$. Observe that $B_{2n}$ has size $4n-4$. The cube $B_{8}$, shown in Figure 3(a), is the smallest biwheel. Except for $B_{8}$, biwheels are noncubic; see Figure 2(b).

On the other hand, prisms and Möbius ladders are cubic; we refer the interested reader to [3, 7] for descriptions of these families. The cube $B_{8}$ is the smallest prism. The smallest Möbius ladders are $K_{3,3}$ and the brace $M_{10}$ shown in Figure 2(a). A McCuaig brace is any brace that is either a prism, or a Möbius ladder, or a biwheel.

1.2 McCuaig’s Theorem

An edge $e$ of a matching covered graph $G$ is removable if $G-e$ is also matching covered. The following is easily deduced from Propositions 1.1 and 1.3.

Corollary 1.6

In a brace, of order six or more, every edge is removable. $\Box$

Now let $G$ denote a brace of order six or more, and let $e\in E(G)$. The bipartite matching covered graph $G-e$ may not be a brace. In particular, one or both ends of $e$ may have degree precisely two (in $G-e$); in order to recover a smaller brace, at the very least, we must get rid of vertices of degree two. This brings us to the following notions of ‘bicontraction’ and ‘retract’.

Let $G$ be a matching covered graph, and let $v_{0}$ denote a vertex of degree two that has two distinct neighbors, say $v_{1}$ and $v_{2}$. The bicontraction of $v_{0}$ is the operation of contracting the two edges $v_{0}v_{1}$ and $v_{0}v_{2}$ incident with $v_{0}$. Note that $\partial(X)$, where $X:=\{v_{0},v_{1},v_{2}\}$, is a tight cut of $G$. The graph obtained by bicontracting $v_{0}$ is the same as $G/X$ and is thus matching covered. However, the bicontraction of a vertex of degree two in a simple graph need not result in a simple graph. The retract of $G$, denoted $\widehat{G}$, is the matching covered graph obtained by bicontracting all its vertices of degree two that have two distinct neighbors.

For a brace $G$ of order six or more, a (removable) edge $e$ is thin if $\widehat{G-e}$ is also a brace. Recently, Carvalho, Lucchesi and Murty [3] proved the following.

Theorem 1.7

Every brace, of order six or more, has at least two thin edges.

Note that if $e$ is a thin edge of a simple brace $G$, the brace $\widehat{G-e}$ may not be simple. A thin edge $e$ of a simple brace $G$ is strictly thin if the brace $\widehat{G-e}$ is also simple. For instance, every edge of $K_{3,3}$, and of $B_{8}$, is thin but none of them is strictly thin. It is easily verified that every McCuaig brace has several thin edges; however, none of them is strictly thin. McCuaig showed that these are in fact the only simple braces with this property. We let $\mathcal{G}$ denote the set that comprises $K_{2}$, $C_{4}$ and all McCuaig braces. McCuaig’s Theorem [7] may now be stated as follows.

Theorem 1.8

[McCuaig’s Theorem]*

Every simple brace $G\notin\mathcal{G}$ has a strictly thin edge.*

Carvalho, Lucchesi and Murty gave an alternative proof of McCuaig’s Theorem using the existence of a thin edge; see [2]. In [3], the same authors establish a stronger version of McCuaig’s Theorem.

For a strictly thin edge $e$ of a simple brace $G$, the index of $e$, denoted ${\rm index}(e)$, is the number of vertices of degree two in $G-e$. Clearly, ${\rm index}(e)\in\{0,1,2\}$, depending on how many ends of $e$ are cubic in $G$. The following is easily verified; see Figures 6 and 7.

Proposition 1.9

Let $e$ denote a strictly thin edge of a simple brace $G$, and let $H:=\widehat{G-e}$. Then $n_{H}=n_{G}-{\rm index}(e)$ and $m_{H}=m_{G}-1-2\cdot{\rm index}(e)$. $\Box$

1.3 Minimal Braces

Recall that a brace $G$ is minimal if, for each $e\in E(G)$, the graph $G-e$ is not a brace. An edge $e$ of a simple brace $G$ is superfluous if $G-e$ is also a brace; note that a ‘superfluous edge’ is the same as a ‘strictly thin edge of index zero’. Thus, a minimal brace is a brace devoid of superfluous edges. Since any superfluous edge must join two noncubic vertices, the following holds.

Proposition 1.10

Let $G$ denote a brace of order six or more. If the set of all noncubic vertices is a stable set then $G$ is a minimal brace. $\Box$

The graph $B_{8}^{+}$ shown in Figure 3(b), obtained from $B_{8}$ by adding an edge, is the smallest simple brace that is not minimal; it has a unique superfluous edge. On the other hand, the graph $Q_{10}^{+}$ shown in Figure 1(b), obtained from $Q_{10}$ by adding an edge, is minimal.

As stated earlier, our main objective is to derive an induction tool for the class of minimal braces from McCuaig’s Theorem.

Now let $G$ denote a minimal brace that is not a member of $\mathcal{G}$. By McCuaig’s Theorem, $G$ has a strictly thin edge, say $e$. We let $H$ denote the simple brace $\widehat{G-e}$. However, $H$ may not be a minimal brace. Clearly, we may choose a set $F\subset E(H)$ such that $J:=H-F$ is a minimal brace. (In this manner, we may recover a smaller minimal brace $J$.) Note that ${\rm index}(e)\in\{1,2\}$, each member of $F$ is a superfluous edge of $H$, and that $F=\emptyset$ if and only if $H$ is a minimal brace. This brings us to the following definition.

Definition 1.11

[Minimality-Preserving Pair]* For a minimal brace $G$, a pair $(e,F)$ is a minimality-preserving pair if $e$ is a strictly thin edge of $G$, and $F$ is a subset of $E(\widehat{G-e})$ so that the graph $\widehat{G-e}-F$ is a minimal brace.*

In the above definition, since each edge of $\widehat{G-e}$ naturally corresponds to an edge of $G$, one may instead view the set $F$ as a subset of $E(G)$. Figure 3 shows an example of a minimal brace $G:=Q_{10}^{+}$ with a minimality-preserving pair $(e,\{f\})$ where ${\rm index}(e)=1$.

The following is a trivial consequence of McCuaig’s Theorem.

Corollary 1.12

Every minimal brace $G\notin\mathcal{G}$ has a minimality-preserving pair $(e,F)$ for any strictly thin edge $e$. $\Box$

The next statement follows immediately from Proposition 1.9.

Proposition 1.13

Let $(e,F)$ denote a minimality-preserving pair of a minimal brace $G$, and let $J:=\widehat{G-e}-F$. Then $n_{J}=n_{G}-{\rm index}(e)$ and $m_{J}=m_{G}-1-2\cdot{\rm index}(e)-|F|$. $\Box$

Corollary 1.12 may be viewed as an induction tool for minimal braces; however, it is not particularly useful for the following reason. If $(e,F)$ is a minimality-preserving pair of a minimal brace $G$, then the minimal brace $J:=\widehat{G-e}-F$ can be arbitrarily smaller in size than $G$ depending on the cardinality of the set $F$. (This is in contrast to McCuaig’s Theorem; see Proposition 1.9). On the other hand, it seems intuitive that for a minimal brace $G\notin\mathcal{G}$ one should be able to find a minimality-preserving pair $(e,F)$ such that the set $F$ is “small”. This is in fact a consequence of our Main Theorem (3.12).

Corollary 1.14

Every minimal brace $G\notin\mathcal{G}$ has a minimality-preserving pair $(e,F)$ such that $|F|\leq{\rm index}(e)+1$.

Apart from this quantitative information regarding the minimality-preserving pair $(e,F)$, our Main Theorem (3.12) also provides qualitative information: for instance, each member of $F$ is at distance one from the strictly thin edge $e$.

Organization of this paper: The Main Theorem (3.12) and its proof appear in Section 3. In Section 4, we use the Main Theorem as an induction tool to prove Theorem 4.2 — which states that $m_{G}\leq 5n_{G}-10$ for any minimal brace $G$, where $n_{G}\geq 6$, and also provides a complete characterization of minimal braces that meet this upper bound. In Section 2, we characterize minimal braces of small order; this will serve as the base case in our proof of Theorem 4.2.

2 Minimal braces of small order

Let $e$ denote any (removable) edge of a simple brace $G$ of order six or more. Note that $e$ is not superfluous if and only if the bipartite matching covered graph $G-e$ has a nontrivial tight cut. One may now easily deduce the following from Proposition 1.2.

Corollary 2.1

Let $G[A,B]$ denote a simple brace of order six or more. An edge $e$ of $G$ is not superfluous if and only if there exist partitions $(A_{1},A_{2})$ of $A$ and $(B_{1},B_{2})$ of $B$ such that $|B_{1}|=|A_{1}|+1$ and $e$ is the only edge joining a vertex in $A_{1}$ to a vertex in $B_{2}$. $\Box$

The brace $Q_{10}^{+}$, shown in Figure 1(b), has precisely one edge joining two noncubic vertices, and it is not superfluous; consequently, $Q_{10}^{+}$ is a minimal brace.

For a simple bipartite graph $G[A,B]$, its bipartite complement $\overline{G}$ is the graph that has the same set of vertices and has edge set $\{ab:ab\notin E(G),a\in A,b\in B\}$. Using this notion, one may easily prove that $K_{3,3}$ and $B_{8}$ are the only simple cubic bipartite graphs of order at most eight. (This proof technique is illustrated in the proof of Proposition 2.3.) Consequently, $K_{3,3}$ is the only simple brace of order six. Using Corollary 2.1, one may infer that $B_{8}$ is the only minimal brace of order eight.

Proposition 2.2

The only minimal braces of order at most eight are $K_{2},C_{4},K_{3,3}$ and $B_{8}$.  $\Box$

Proposition 2.3

The only simple cubic bipartite graphs of order ten are $M_{10}$ and $Q_{10}$.

Proof: Let $G[A,B]$ denote a simple cubic bipartite graph of order ten, and let $\overline{G}$ denote its bipartite complement. Clearly, $\overline{G}$ is $2$-regular. Observe the following. If $\overline{G}$ is disconnected then $G$ is isomorphic to $Q_{10}$. Otherwise $\overline{G}$ is connected and $G$ is isomorphic to $M_{10}$. $\Box$

Proposition 2.4

The only minimal braces of order ten are $M_{10}$, $B_{10}$ and $Q_{10}^{+}$.

Proof: Let $G[A,B]$ denote a minimal brace of order ten. If $G$ is cubic then, by Proposition 2.3, $G$ is isomorphic to $M_{10}$. Now assume that $G$ is noncubic, and let $T$ denote the set of noncubic vertices.

First suppose that $T$ is a stable set. Observe that $T$ has precisely two members, say $a$ and $b$, each of which has degree precisely four. Consequently, $G-a-b$ is a connected $2$-regular bipartite graph. (Recall that braces of order six or more are $3$-connected.) Thus $G-a-b$ is isomorphic to $C_{8}$ and $G$ is isomorphic to $B_{10}$.

Now suppose that $T$ is not a stable set, and let $e:=ab$ denote an edge that joins $a\in A\cap T$ and $b\in B\cap T$. Since $G$ is devoid of superfluous edges, the edge $e$ in particular is not superfluous. By Corollary 2.1, there exist partitions $(A_{1},A_{2})$ of $A$ and $(B_{1},B_{2})$ of $B$ such that $|B_{1}|=|A_{1}|+1$ and $e$ is the only edge joining a vertex in $A_{1}$ to a vertex in $B_{2}$. Since $a$ and $b$ are noncubic vertices, we infer that each of the sets $B_{1}$ and $A_{2}$ has at least three vertices. Consequently, $|B_{1}|=|A_{2}|=3$, and each of the induced subgraphs $G[A_{1}\cup B_{1}]$ and $G[A_{2}\cup B_{2}]$ is isomorphic to $K_{2,3}$. Since $|A_{1}|=|B_{2}|=2$, by Proposition 1.3, the induced subgraph $G[A_{2}\cup B_{1}]$ has a perfect matching. All of these facts imply that the brace $Q_{10}^{+}$ is a subgraph of $G$, whence $G$ is isomorphic to $Q_{10}^{+}$. $\Box$

Definition 2.5

[Stable-Extension]* Let $S$ denote a stable set of a connected bipartite graph $J[A,B]$ that meets each color class in precisely two vertices. We let $S:=\{a_{1},a_{2},b_{1},b_{2}\}$ where $a_{1},a_{2}\in A$ and $b_{1},b_{2}\in B$. The graph $G$ obtained from $J$ — by adding two new vertices $a_{0}$ and $b_{0}$, and five new edges $a_{0}b_{1},a_{0}b_{2},b_{0}a_{1},b_{0}a_{2}$ and $a_{0}b_{0}$ — is called the stable-extension of $J$ with respect to $S$ — or simply the $S$-extension of $J$. We refer to $a_{0}$ and $b_{0}$ as the extension vertices of $G$.*

The graph $Q_{10}$, shown in Figure 4(a), has a unique stable set $S$ that meets each color class in precisely two vertices. We let $Q_{12}$, shown in Figure 4(b), denote the $S$-extension of $Q_{10}$. Using Propositions 1.3 and 1.10, one may verify that $Q_{12}$ is a minimal brace.

Proposition 2.6

The only minimal braces of order $12$, and size at least $20$, are $B_{12}$ and $Q_{12}$.

Proof: Let $G[A,B]$ denote a minimal brace of order $12$ and size at least $20$, and let $T$ denote the set of noncubic vertices. Clearly, each of the sets $T\cap A$ and $T\cap B$ is nonempty. We let $E_{T}$ denote the set of edges that have both ends in $T$.

First suppose that $T$ is a stable set (i.e., $E_{T}=\emptyset$), whence $G$ has maximum degree at most five. Observe that if there exists a vertex of degree five then $|T\cap A|=|T\cap B|=1$; whence $G-T$ is a connected $2$-regular bipartite graph; consequently, $G-T$ is isomorphic to $C_{10}$ and $G$ is isomorphic to $B_{12}$. Otherwise, $|T\cap A|=|T\cap B|=2$, and each member of $T$ has degree precisely four. In this case, observe that $G-T$ is a $1$-regular (bipartite) graph; thus, $G-T$ has four components (each isomorphic to $K_{2}$), and $G$ is isomorphic to $Q_{12}$.

Now suppose that $T$ is not a stable set. Our goal is to arrive at a contradiction; however, it requires some tedious arguments.

Let $e:=ab$ denote any member of $E_{T}$. Since $e$ is not superfluous, by Corollary 2.1, there exist partitions $(A_{1},A_{2})$ of $A$ and $(B_{1},B_{2})$ of $B$ such that $|B_{1}|=|A_{1}|+1$ and $e$ is the only edge joining a vertex in $A_{1}$ to a vertex in $B_{2}$. Since each end of $e$ is noncubic, and since $n_{G}=6$, one of the two sets $B_{1}$ and $A_{2}$ has cardinality three and the other one has cardinality four. Adjust notation so that $|B_{1}|=3$ and $|A_{2}|=4$. See Figure 5(a). Let $a_{1}$ denote the unique member of $A_{1}-a$. Observe that $N_{G}(a)=B_{1}\cup\{b\}$ and $N_{G}(a_{1})=B_{1}$.

In particular, we have proved the following.

2.6.1

Each edge $f\in E_{T}$ has an end whose degree is precisely four, say $v$, such that there exists a cubic vertex $v^{\prime}$ that satisfies $N_{G}(v^{\prime})=N_{G-f}(v)$. $\Box$

The degree of vertex $b$ is either four or five; we will prove that it must be four. Suppose to the contrary that $b$ has degree five; whence $N_{G}(b):=A_{2}\cup\{a\}$. By a simple counting argument, there exists a noncubic vertex, say $a_{2}$, in $A_{2}$. Since $f_{2}:=a_{2}b$ is a member of $E_{T}$, using statement 2.6.1, we infer that $a_{2}$ has degree precisely four, and there exists a cubic vertex $a^{\prime}$ that satisfies $N_{G}(a^{\prime})=N_{G-f_{2}}(a_{2})=N_{G}(a_{2})-b$. Since each member of $A_{2}$ is adjacent with $b$, it follows that $a^{\prime}=a_{1}$. Consequently, $N_{G}(a_{2})=B_{1}\cup\{b\}$. This implies that $N_{G}(B_{2}-b)\subseteq A_{2}-a_{2}$. This contradicts Proposition 1.3. Thus $b$ has degree four.

In particular, we have now proved the following.

2.6.2

For every edge $f\in E_{T}$, each end of $f$ has degree precisely four. (Consequently, $|T\cap A|=|T\cap B|\geq 2$, and each vertex in $T$ has degree precisely four.) $\Box$

Now, we will prove that each neighbor of $a$, distinct from $b$, is cubic. Suppose to the contrary that there exists $b_{1}\in B_{1}$ that is noncubic. Thus $b_{1}$ has degree precisely four and $f_{1}:=ab_{1}$ is a member of $E_{T}$. Now we invoke statement 2.6.1. Either there exists a cubic vertex $a^{\prime}$ that satisfies $N_{G}(a^{\prime})=N_{G-f_{1}}(a)=(B_{1}-b_{1})\cup\{b\}$, or otherwise there exists a cubic vertex $b^{\prime}$ that satisfies $N_{G}(b^{\prime})=N_{G-f_{1}}(b_{1})=N_{G}(b_{1})-a$. In the latter case, note that $b^{\prime}\in B_{2}$ (since each vertex in $B_{1}$ is adjacent with $a$); however, this implies that we have an edge joining $b^{\prime}\in B_{2}$ and $a_{1}\in A_{1}$; contradiction. In the former case, note that $a^{\prime}\in A_{2}$; whence $N_{G}(B_{2}-b)\subseteq A_{2}-a^{\prime}$, and this contradicts Proposition 1.3. Thus each neighbor of $a$, distinct from $b$, is cubic; see Figure 5(b).

In particular, we have established the following.

2.6.3

For every edge $f\in E_{T}$, each end of $f$ has degree precisely four (in $G$) and at least one of them has degree precisely one in the induced subgraph $G[T]$. $\Box$

We let $w\in A$ and $u\in B$ denote two vertices of $T$ that are distinct from $a$ and $b$. Note that $w\in A_{2}$ and $u\in B_{2}$. By Proposition 1.3, the graph $G-a-a_{1}-b-u$ has a perfect matching; whence the three edges joining $B_{1}$ and $A_{2}$ constitute a matching. This implies that $w$ has precisely one neighbor in $B_{1}$. Consequently, $wb,wu\in E(G)$. Observe that the induced subgraph $G[T]$ contains a path of length three: $(a,b,w,u)$. The edge $bw\in E_{T}$ contradicts statement 2.6.3.

This completes the proof of Proposition 2.6. $\Box$

3 An induction tool for minimal braces

In subsection 1.2, we presented a ‘reduction version’ of McCuaig’s Theorem (1.8) using the notion of a strictly thin edge; this viewpoint and the associated terminology is due to Carvalho, Lucchesi and Murty [2, 3] and it is convenient for stating results concisely. In the following subsection, we shall present a ‘generation version’ of McCuaig’s Theorem (3.4). In order to do so, we first need to define some ‘expansion operations’; these will also be useful in deducing our Main Theorem (3.12).

3.1 Expansion operations

For a simple bipartite connected graph $H[A,B]$, and nonadjacent vertices $a\in A$ and $b\in B$, $H+ab$ denotes the graph obtained from $H$ by adding the edge $ab$. Note that if $H$ is a brace then, by Corollary 1.4, $H+ab$ is a (simple) brace and $ab$ is a strictly thin edge of index zero; in this case, we say that $H+ab$ is obtained from $H$ by an expansion of index zero. We shall now define two more expansion operations (on simple braces) — each of which may be viewed as the reverse of removing a strictly thin edge (of index one or two) and then taking the retract. To do so, we first need the notion of ‘bi-splitting’ a noncubic vertex.

Let $b$ denote a noncubic vertex of a simple bipartite matching covered graph $H[A^{\prime},B^{\prime}]$. Adjust notation so that $b\in B^{\prime}$. Suppose that a (bipartite) graph $G$ is obtained from $H$ by replacing the vertex $b$ by two new vertices $b_{1}$ and $b_{2}$, distributing the edges in $H$ incident with $b$ between $b_{1}$ and $b_{2}$ such that each gets at least two edges, and then adding a new vertex $a_{0}$ and two new edges: $a_{0}b_{1},a_{0}b_{2}$. We say that $G$ is obtained from $H$ by bi-splitting $b$ into $b_{1}a_{0}b_{2}$, and we denote this as $G:=H\{b\rightarrow b_{1}a_{0}b_{2}\}$. It is easily verified that $G$ is also matching covered. Observe that $H$ can be recovered from $G$ by bicontracting the vertex $a_{0}$ and denoting the contraction vertex by $b$.

We are now ready to define the aforementioned expansion operations. (See Figures 6 and 7.)

Definition 3.1

[Expansion of Index One]* Choose two vertices, say $a$ and $w$, of a simple brace $H$ that belong to the same color class such that at least one of them, say $a$, is noncubic. A graph $G$ is obtained from $H$ by an expansion of index one if $G:=H\{a\rightarrow a_{1}b_{0}a_{2}\}+b_{0}w$.*

As an example, one may construct the brace $Q_{12}$ from the brace $Q_{10}^{+}$ by means of an expansion of index zero (i.e., adding an edge) followed by an expansion of index one. To see this, we let $Q_{10}^{+}:=Q_{10}+a_{1}b_{1}$ as per the labeling in Figure 4(a). Now, observe that, as per the labeling in Figure 4(b), $Q_{12}:=(Q_{10}^{+}+a_{1}b_{2})\{a_{1}\rightarrow a_{1}b_{0}a_{0}\}+b_{0}a_{2}$.333In order to be consistent with the labeling used in Figures 4(a) and 4(b), we slightly abuse notation by overloading the label $a_{1}$.

Definition 3.2

[Expansion of Index Two]* Choose two noncubic vertices, say $a$ and $b$, of a simple brace $H$ that lie in distinct color classes. A graph $G$ is obtained from $H$ by an expansion of index two if $G:=H\{a\rightarrow a_{1}b_{0}a_{2}\}\{b\rightarrow b_{1}a_{0}b_{2}\}+a_{0}b_{0}$.*

For convenience, we say that a graph $G$ is obtained from the simple brace $H$ by an expansion operation if $G$ is obtained from $H$ by an expansion of index zero, one or two. McCuaig [7] proved the following.

Proposition 3.3

Any graph $G$, that is obtained from a simple brace $H$ by an expansion operation, is also a simple brace.

Let $H[A,B]$ denote a simple brace. Observe the following. If $G:=H\{a\rightarrow a_{1}b_{0}a_{2}\}+b_{0}w$ for some $a,w\in A$, then $e:=b_{0}w$ is a strictly thin edge of index one. On the other hand, if $G:=H\{a\rightarrow a_{1}b_{0}a_{2}\}\{b\rightarrow b_{1}a_{0}b_{2}\}+a_{0}b_{0}$ for some $a\in A$ and $b\in B$, then $e:=a_{0}b_{0}$ is a strictly thin edge of index two. In either case, $H$ is isomorphic to $\widehat{G-e}$. We now state the ‘generation version’ of McCuaig’s Theorem (that is equivalent to Theorem 1.8.)

Theorem 3.4

Every simple brace $G\notin\mathcal{G}$ may be obtained from a (smaller) simple brace by an expansion operation.

In the following two subsections, we state and prove several lemmas; these will culminate in the proof of the Main Theorem (3.12) that appears in subsection 3.4. However, we need some more terminology in order to state these lemmas.

Consider the graph $G-e$ where $e$ is a strictly thin edge of a simple brace $G$. In $G-e$, a vertex of degree two is referred to as an inner vertex, and each of its neighbors is referred to as an outer vertex. Note that $G-e$ has precisely ${\rm index}(e)$ inner vertices and $2\cdot{\rm index}(e)$ outer vertices; see Figures 6 and 7.

3.2 Index one

Throughout this subsection, we assume that $G[A,B]\notin\mathcal{G}$ is a minimal brace, and that $e:=b_{0}w$ is a strictly thin edge of index one where $b_{0}\in B$ is the cubic end of $e$. We also assume that the simple brace $H:=\widehat{G-e}$ is not minimal; whence $H$ has superfluous edge(s). Thus $n_{H}\geq 4$ and $n_{G}\geq 5$. We let $a_{1}$ and $a_{2}$ denote the outer vertices of $G-e$. See Figure 6.

Lemma 3.5

Let $f$ denote any superfluous edge of $H$. Then $G-e$ has a cubic outer vertex that is an end of $f$, whereas the other end of $f$ is noncubic.

Proof: Recall definition 3.1, and observe the following. If each of $a_{1}$ and $a_{2}$ has degree at least three in $G-f$, then $G-f$ may be obtained from the brace $H-f$ by an expansion of index one; whence $G-f$ is a brace (by Proposition 3.3); contradiction. It follows that one of $a_{1}$ and $a_{2}$ is a cubic end of $f$. Clearly, the other end of $f$ is noncubic. $\Box$

Lemma 3.6

Let $f_{1}$ and $f_{2}$ denote two edges of $H$ such that $H-f_{1}-f_{2}$ is a brace. Then $G-e$ has a cubic outer vertex that is a common end of $f_{1}$ and $f_{2}$.

Proof: It follows from the hypothesis that each of $f_{1}$ and $f_{2}$ is a superfluous edge of $H$. By Lemma 3.5, for $i\in\{1,2\}$, the edge $f_{i}$ has precisely one cubic end (in $G$) that lies in $\{a_{1},a_{2}\}$. If the cubic end of $f_{1}$ is the same as the cubic end of $f_{2}$, then there is nothing to prove.

Now suppose that $f_{i}\in\partial(a_{i})$ for $i\in\{1,2\}$. Consequently, $a_{1}$ and $a_{2}$ are both cubic (in $G$), whence $a$ has degree precisely four in $H$; consequently, $a$ is a vertex of degree two in $H-f_{1}-f_{2}$; contradiction. $\Box$

Note that, if $(e,F)$ is a minimality-preserving pair of $G$ then any two distinct members of $F$, say $f_{1}$ and $f_{2}$, satisfy the hypothesis of Lemma 3.6. We thus have the following immediate consequence.

Corollary 3.7

If $(e,F)$ is a minimality-preserving pair of $G$, then $G-e$ has a cubic outer vertex $v$ such that $F\subset\partial(v)$. Consequently, $|F|\in\{1,2\}$. $\Box$

Lemma 3.8

Suppose that $(e,F)$ is a minimality-preserving pair of $G$ such that $|F|=2$. Then, for each $f\in F$, the noncubic end of $f$ is not adjacent with the noncubic end of $e$. Moreover, $G$ is isomorphic to a stable-extension of the minimal brace $J:=\widehat{G-e}-F$.

Proof: Let $F:=\{f_{1},f_{2}\}$. We invoke Lemma 3.6, and adjust notation so that $a_{1}$ is cubic and $f_{1},f_{2}\in\partial(a_{1})$. We let $u_{1}$ and $u_{2}$ denote the noncubic ends of $f_{1}$ and $f_{2}$, respectively. We let $J:=H-F$. Thus $J$ is a minimal brace. See Figure 8.

Our goal is to prove that $wu_{1},wu_{2}\notin E(G)$. By symmetry, it suffices to prove that $wu_{2}\notin E(G)$. Suppose to the contrary that $wu_{2}\in E(G)$. See Figure 9.

In this case, $w$ and $u_{2}$ are adjacent in $J$ as well. Observe that one may obtain the graph $G^{\prime}:=G-wu_{2}$ from the brace $J^{\prime}:=J+au_{2}$ by an expansion of index one. In particular, $G^{\prime}=J^{\prime}\{u_{2}\rightarrow u_{2}a_{1}b_{0}\}+a_{1}u_{1}$. By Proposition 3.3, $G^{\prime}$ is a brace — contrary to our hypothesis that $G$ is minimal.

We thus conclude that $wu_{1},wu_{2}\notin E(G)$. Consequently, $S:=\{a,w,u_{1},u_{2}\}$ is a stable set of $J$ that meets each color class in two vertices. Observe that $G$ is isomorphic to the $S$-extension of $J$ (with $a_{1}$ and $b_{0}$ playing the role of the extension vertices). This completes the proof of Lemma 3.8. $\Box$

3.3 Index two

Throughout this subsection, we assume that $G[A,B]\notin\mathcal{G}$ is a minimal brace that is devoid of strictly thin edges of index one, and that $e:=a_{0}b_{0}$ is a strictly thin edge of index two. We also assume that the simple brace $H:=\widehat{G-e}$ is not minimal; whence $H$ has superfluous edge(s). Thus $n_{H}\geq 4$ and $n_{G}\geq 6$. We let $a_{1},a_{2}\in A$ and $b_{1},b_{2}\in B$ denote the outer vertices of $G-e$. See Figure 7.

Since $e$ is a strictly thin edge, $G$ has at most one edge that has one end in $\{a_{1},a_{2}\}$ and the other end in $\{b_{1},b_{2}\}$; furthermore, $G$ has (precisely) one such edge if and only if $ab\in E(H)$.

Lemma 3.9

Let $f$ denote any superfluous edge of $H$. Then $G-e$ has a cubic outer vertex that is an end of $f$; the other end of $f$ is either noncubic, or it is another cubic outer vertex.

Proof: Recall definition 3.2, and observe the following. If each of $a_{1},a_{2},b_{1},b_{2}$ has degree at least three in $G-f$, then $G-f$ may be obtained from the brace $H-f$ by an expansion of index two; whence $G-f$ is a brace (by Proposition 3.3); contradiction. It follows that one of $a_{1},a_{2},b_{1},b_{2}$ is a cubic end of $f$. Adjust notation so that $a_{1}$ is a cubic end of $f$. Clearly, if the other end of $f$ is not in $\{b_{1},b_{2}\}$ then it is noncubic.

Now suppose that the other end of $f$ is in $\{b_{1},b_{2}\}$; adjust notation so that $f:=a_{1}b_{1}$. Assume that $b_{1}$ is noncubic. Let $u$ denote the neighbor of $a_{1}$ that is distinct from $b_{0}$ and $b_{1}$. See Figure 10. Observe that the graph $\widehat{G-f}$ may be obtained from the brace $H-f$ by an expansion of index one; in particular, $\widehat{G-f}:=(H-f)\{b\rightarrow b_{1}a_{0}b_{2}\}+a_{0}u$. Thus $\widehat{G-f}$ is a brace (by Proposition 3.3). Consequently, $f$ is a strictly thin edge of index one, contrary to our hypothesis that $G$ is devoid of such edges. Thus $b_{1}$ is also cubic. $\Box$

Lemma 3.10

Let $f$ denote any superfluous edge of $H$. Then, in $G-e$, either $f$ joins two outer vertices, or otherwise $f$ is adjacent with an edge that joins two outer vertices.

Proof: By Lemma 3.9, $G-e$ has a cubic outer vertex that is an end of $f$. Adjust notation so that $a_{1}$ is a cubic end of $f$. If the other end of $f$ lies in $\{b_{1},b_{2}\}$ then we are done.

Now suppose that $f\notin\partial(\{b_{1},b_{2}\})$, and let $u^{\prime}$ denote the noncubic end of $f$. Assume that $a_{1}b_{1},a_{1}b_{2}\notin E(G)$. Let $u$ denote the neighbor of $a_{1}$ that is distinct from $b_{0}$ and $u^{\prime}$. Note that $u\notin\{b_{1},b_{2}\}$. See Figure 11. Observe that the graph $\widehat{G-f}$ may be obtained from the brace $H-f$ by an expansion of index one; in particular, $\widehat{G-f}:=(H-f)\{b\rightarrow b_{1}a_{0}b_{2}\}+a_{0}u$. Thus $\widehat{G-f}$ is a brace (by Proposition 3.3). Consequently, $f$ is a strictly thin edge of index one, contrary to our hypothesis that $G$ is devoid of such edges. Thus, one of $a_{1}b_{1}$ and $a_{1}b_{2}$ is an edge of $G$; whence $f$ is adjacent with an edge that joins two outer vertices. $\Box$

Corollary 3.11

If $(e,F)$ is a minimality-preserving pair of $G$, then $G-e$ has two adjacent outer vertices $u$ and $w$ such that $F\subset\partial(u)\cup\partial(w)$. Furthermore, if $F\cap\partial(u)\neq\emptyset$ then $u$ is cubic; likewise, if $F\cap\partial(w)\neq\emptyset$ then $w$ is cubic. Consequently, $|F|\in\{1,2,3\}$. $\Box$

3.4 Main Theorem

We are now ready to state and prove the Main Theorem.

Theorem 3.12

[Main Theorem]* Every minimal brace $G\notin\mathcal{G}$ has a minimality-preserving pair $(e,F)$ such that either $F=\emptyset$, or otherwise $F$ satisfies the following properties in $G-e$:*

1. (i)

If ${\rm index}(e)=1$ then $G-e$ has an outer vertex, say $v$, such that $F\subset\partial(v)$. 2. (ii)

If ${\rm index}(e)=2$ then $G-e$ has two adjacent outer vertices, say $u$ and $w$, such that $F\subset\partial(u)\cup\partial(w)$. 3. (iii)

For each $f\in F$, an end of $f$ is cubic if and only if it is an outer vertex.

Consequently, $|F|\leq{\rm index}(e)+1$. Furthermore, if ${\rm index}(e)=1$ and $|F|=2$ then $G$ is isomorphic to a stable-extension of the minimal brace $J:=\widehat{G-e}-F$.

Proof: By McCuaig’s Theorem and its corollary (1.12), $G$ has a strictly thin edge (of index one or two); furthermore, for any such edge $e$, there exists a minimality-preserving pair $(e,F)$.

If $G$ has a strictly thin edge of index one, say $e$, then we choose any minimality-preserving pair $(e,F)$, and we are done by invoking Lemma 3.5, Corollary 3.7 and Lemma 3.8.

Now suppose that $G$ is devoid of strictly thin edges of index one. Thus $G$ has a strictly thin edge of index two, say $e$. We choose any minimality-preserving pair $(e,F)$, and we are done by invoking Lemma 3.9 and Corollary 3.11. $\Box$

For convenience, let us introduce the following definition.

Definition 3.13

[Narrow Minimality-Preserving Pair]* A minimality-preserving pair $(e,F)$ of a minimal brace $G$ is narrow if it satisfies the statement of the Main Theorem (3.12).*

We may thus condense the statement of the Main Theorem as follows: every minimal brace $G\notin\mathcal{G}$ has a narrow minimality-preserving pair.

Recall that we presented two equivalent versions of McCuaig’s Theorem: first a ‘reduction version’ (1.8), and second a ‘generation version’ (3.4). In the same spirit, Theorem 3.12 (as stated) may be viewed as a ‘reduction version’ for minimal braces. The interested reader may obtain a ‘generation version’ (equivalent to Theorem 3.12) for minimal braces by defining ‘extension operations’ that could be potentially useful in obtaining a (larger) minimal brace from a (smaller) minimal brace. Each such extension operation may be viewed as a sequence of index zero expansions (possibly none) followed by an index one or index two expansion with some additional restrictions (depending on the cardinality and structure of the set $F$ and the index of the strictly thin edge $e$). This is in fact the viewpoint adopted by Norine and Thomas [10] in their paper on minimal bricks.

We shall not define all of the aforementioned extension operations here (except for the one we have already described: Definition 2.5). The reader may verify that if a graph $G$ is isomorphic to a stable-extension of a simple brace $J$, then $G$ may be obtained from $J$ by first adding two (adjacent) edges and then performing an index one expansion. This observation, coupled with Proposition 3.3, yields the following.

Corollary 3.14

Any graph $G$, that is isomorphic to a stable-extension of a simple brace $J$, is also a simple brace. $\Box$

4 An application

In this section, as an application of the Main Theorem (3.12), we will prove that if $G$ is a minimal brace (where $n_{G}\geq 6$) then $m_{G}\leq 5\cdot n_{G}-10$; furthermore, we shall provide a complete characterization of minimal braces that meet this upper bound.

We begin by defining an infinite family $\mathcal{Q}$ of minimal braces, each of whose members meets this upper bound; its smallest member is $Q_{12}$. For $n\geq 7$, we define the graph $Q_{2n}$ as the $S$-extension of $Q_{2n-2}$ — where the set $S$ comprises the noncubic vertices of $Q_{2n-2}$. Now let $\mathcal{Q}:=\{Q_{2n}:n\geq 6\}$. Since $Q_{12}$ is a simple brace, it follows from Corollary 3.14 that each member of $\mathcal{Q}$ is a simple brace. Furthermore, by Proposition 1.10, we infer that each member of $\mathcal{Q}$ is in fact a minimal brace. By summing degrees, observe that if $G\in\mathcal{Q}$ then $m_{G}=5\cdot n_{G}-10$.

We now prove the following easy corollary of the Main Theorem.

Corollary 4.1

Let $(e,F)$ denote a narrow minimality-preserving pair of a minimal brace $G$, and let $J:=\widehat{G-e}-F$. Assume that $m_{J}\leq 5\cdot n_{J}-10$. Then the following hold:

1. (i)

$m_{G}\leq 5\cdot n_{G}-10$. 2. (ii)

If $m_{G}=5\cdot n_{G}-10$ then $m_{J}=5\cdot n_{J}-10$, ${\rm index}(e)=1$ and $|F|=2$; consequently, $G$ is isomorphic to a stable-extension of $J$.

Proof: By Proposition 1.13, $n_{J}=n_{G}-{\rm index}(e)$ and $m_{G}=m_{J}+1+2\cdot{\rm index}(e)+|F|$. By Theorem 3.12, $|F|\leq{\rm index}(e)+1$. We now consider two cases depending on the index of $e$.

First consider the case: ${\rm index}(e)=2$. Using the equations and inequalities noted above, we have: $m_{G}=m_{J}+5+|F|<(5\cdot n_{J}-10)+5+5=5\cdot n_{J}=5\cdot n_{G}-10.$ Thus, in this case, the strict inequality $m_{G}<5\cdot n_{G}-10$ holds.

Now consider the case: ${\rm index}(e)=1$. Using the same equations and inequalities as before, we have: $m_{G}=m_{J}+3+|F|\leq(5\cdot n_{J}-10)+3+2=5\cdot n_{J}-5=5\cdot n_{G}-10$. Hence, we have the inequality $m_{G}\leq 5\cdot n_{G}-10$; equality holds if and only if $m_{J}=5\cdot n_{J}-10$ and $|F|=2$. By the last part of Theorem 3.12, we infer that $G$ is isomorphic to a stable-extension of $J$. This completes the proof of Corollary 4.1. $\Box$

We are now ready to prove the main result of this section.

Theorem 4.2

Let $G$ denote a minimal brace that is not in $\{K_{2},C_{4},K_{3,3},B_{8},B_{10},Q_{10}^{+}\}$. Then $m_{G}\leq 5\cdot n_{G}-10$, and equality holds if and only if $G\in\{M_{10},B_{12}\}\cup\mathcal{Q}$.

Proof: We proceed by induction on the order of the graph. Let $G[A,B]$ denote a minimal brace that is not in $\{K_{2},C_{4},K_{3,3},B_{8},B_{10},Q_{10}^{+}\}$.

The reader may verify, using Propositions 2.2 and 2.4, that the desired conclusion holds when $n_{G}\leq 5$. Now suppose that $n_{G}\geq 6$. If $G$ is cubic then $m_{G}=3\cdot n_{G}<5\cdot n_{G}-10$. We may thus assume that $G$ is noncubic. Now one may verify, using Proposition 2.6, that the desired conclusion holds when $n_{G}=6$. Henceforth, suppose that $n_{G}\geq 7$. If $G$ is a biwheel then $m_{G}=4\cdot n_{G}-4<5\cdot n_{G}-10$. We may thus assume that $G$ is not a biwheel. Consequently, $G\notin\mathcal{G}$.

By the Main Theorem (3.12), $G$ has a narrow minimality-preserving pair, say $(e,F)$. Let $J$ denote the minimal brace $\widehat{G-e}-F$. Note that $n_{J}=n_{G}-{\rm index}(e)$ where ${\rm index}(e)\in\{1,2\}$; whence $n_{J}\geq 5$. By invoking the induction hypothesis, either $J\in\{B_{10},Q_{10}^{+}\}$, or otherwise $m_{J}\leq 5\cdot n_{J}-10$ and equality holds if and only if $J\in\{M_{10},B_{12}\}\cup\mathcal{Q}$.

First suppose that $J\in\{B_{10},Q_{10}^{+}\}$. In particular, $n_{J}=5$ and $m_{J}=16$. Consequently, ${\rm index}(e)=2$ and $n_{G}=7$. By Proposition 1.13, $m_{G}=m_{J}+1+2\cdot{\rm index}(e)+|F|$ where $|F|\leq 3$. It follows that $m_{G}\leq 24<25=5\cdot n_{G}-10$. The desired conclusion holds.

Now suppose that $m_{J}\leq 5\cdot n_{J}-10$ and equality holds if and only if $J\in\{M_{10},B_{12}\}\cup\mathcal{Q}$. By invoking Corollary 4.1, we infer that $m_{G}\leq 5\cdot n_{G}-10$. Since $n_{G}\geq 7$, it only remains to prove the following: if $m_{G}=5\cdot n_{G}-10$ then $G\in\mathcal{Q}$.

Henceforth, assume that $m_{G}=5\cdot n_{G}-10$. It now follows from Corollary 4.1 that $m_{J}=5\cdot n_{J}-10$. Thus $J\in\{M_{10},B_{12}\}\cup\mathcal{Q}$. Furthermore, $J$ has a stable set $S$ that meets each color class in precisely two vertices, and $G$ is isomorphic to the $S$-extension of $J$. We let $A^{\prime}$ and $B^{\prime}$ denote the color classes of $J$, and we let $a_{0}$ and $b_{0}$ denote the extension vertices of $G$. Thus $A=A^{\prime}\cup\{a_{0}\}$ and $B=B^{\prime}\cup\{b_{0}\}$.

One may easily verify that $M_{10}$, shown in Figure 2(a), has no stable set that meets each color class in two vertices. Thus, either $J$ is isomorphic to $B_{12}$ or otherwise $J\in\mathcal{Q}$.

First consider the case in which $J$ is isomorphic to $B_{12}$. We let $a\in A^{\prime}$ and $b\in B^{\prime}$ denote the noncubic vertices of $J$. The reader may easily verify that, up to symmetry, $B_{12}$ has only one stable set $\{a_{1},b_{1},a_{2},b_{2}\}$ that meets each color class in precisely two vertices — as shown in Figure 12(a). Thus, we let $S:=\{a_{1},b_{1},a_{2},b_{2}\}$; whence $G$ is isomorphic to the graph shown in Figure 12(b). We shall now arrive at a contradiction by showing that $G-f$ is a brace — where $f:=a_{1}b$. By Proposition 1.3, for each nonempty set $Z\subset A$ such that $|Z|<|A|-1$, the inequality $|N_{G}(Z)|\geq|Z|+2$ holds; furthermore, it suffices to verify that $|N_{G-f}(Z)|\geq|Z|+2$ holds; by comparing the graphs $G$ and $G-f$, observe that we only need to check those sets $Z$ that satisfy the following: $\{a_{1}\}\subseteq Z\subseteq\{a,a_{0},a_{1}\}$. The reader may verify that, for each set $Z$ that satisfies $\{a_{1}\}\subseteq Z\subseteq\{a,a_{0},a_{1}\}$, the inequality $|N_{G-f}(Z)|\geq|Z|+2$ holds. Thus $f$ is a superfluous edge in $G$; contradiction. (In fact, by symmetry, any edge of $G$, both of whose ends are noncubic, is superfluous.)

Now consider the case in which $J\in\mathcal{Q}$. We let $A^{\prime}:=\{a_{1},a_{2},w_{1},w_{2},\dots,w_{n_{J}-2}\}$ and $B^{\prime}:=\{b_{1},b_{2},u_{1},u_{2},\dots,u_{n_{J}-2}\}$ — such that $T:=\{a_{1},a_{2},b_{1},b_{2}\}$ is the set of noncubic vertices of $J$, and $w_{i}u_{i}\in E(J)$ for each $i\in\{1,2,\dots,n_{J}-2\}$ — as shown in Figure 13(a). We consider two subcases depending on whether the set $T$ intersects with the stable set $S$ or not.

First suppose that $S\cap T=\emptyset$. Thus, we may adjust notation so that $S:=\{w_{1},w_{2},u_{3},u_{4}\}$; see Figure 13(b). Similarly to an earlier case, we will arrive at a contradiction by showing that $G-f$ is a brace — where $f:=b_{1}w_{1}$. By Proposition 1.3, for each nonempty set $Z\subset A$ such that $|Z|<|A|-1$, the inequality $|N_{G}(Z)|\geq|Z|+2$ holds; furthermore, it suffices to verify that $|N_{G-f}(Z)|\geq|Z|+2$ holds; by comparing the graphs $G$ and $G-f$, observe that we only need to check those sets $Z$ that satisfy the following: $\{w_{1}\}\subseteq Z\subseteq\{w_{1},a_{0},a_{1},a_{2}\}$. The reader may verify that, for each set $Z$ that satisfies $\{w_{1}\}\subseteq Z\subseteq\{w_{1},a_{0},a_{1},a_{2}\}$, the inequality $|N_{G-f}(Z)|\geq|Z|+2$ holds. Thus $f$ is a superfluous edge in $G$; contradiction. (In fact, by symmetry, any edge of $G$, both of whose ends are noncubic, is superfluous.)

Now suppose that $S\cap T\neq\emptyset$. Observe that, in this case, $S=T$. Consequently, $G\in\mathcal{Q}$, by definition of the family $\mathcal{Q}$. This completes the proof of Theorem 4.2. $\Box$

Acknowledgements: The first and third authors received support from Fundect-MS and from CNPq (Brazil). The second author received support from FAPESP (2018/04679-1) of Sao Paulo (Brazil) and from the Austrian Science Foundation FWF (START grant Y463). The authors are especially grateful to one of the anonymous referees who helped in improving the presentation.