Minimal Braces
Phelipe A. Fabres, Nishad Kothari, Marcelo H. de Carvalho

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 has size at most , when , 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…
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Taxonomy
TopicsComputational Geometry and Mesh Generation · Advanced Graph Theory Research · semigroups and automata theory
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Minimal Braces111This research is supported by
Fundect-MS, CNPq and FAPESP (2018/04679-1) of Brazil, and by the Austrian Science Foundation FWF (START grant Y463).
Phelipe A. Fabresa
Nishad Kotharib
Marcelo H. de Carvalhoa
a FACOM — UFMS, Campo Grande, Brasil
b IC — UNICAMP, Campinas, Brasil
(1 September, 2020)
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, Pólya’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 has size at most , when , 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 , its order is the number of vertices (i.e., ), and its size is the number of edges (i.e., ). For a subset of , we denote by the cut associated with , and we refer to and as the shores of . Thus 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 to a single vertex is denoted by , or simply by . The two graphs and are called the -contractions of .
A connected graph is -extendable if it has a matching of cardinality , and if each such matching extends to (i.e., is a subset of) a perfect matching of . 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 -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 -connected. Also, for a graph , we let and ; whence has order and size .
A cut of is tight if for each perfect matching of . 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 is a nontrivial tight cut of a matching covered graph then each -contraction of 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 . 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 yield the same list of bricks and braces (except possibly for the multiplicities of edges). We remark that is bipartite if and only if its tight cut decomposition yields only braces (i.e., it yields no bricks). The bipartite matching covered graph , shown in Figure 1(a), has a nontrivial tight cut , and each of its -contractions is isomorphic to the brace .
Several important properties of a matching covered graph may be deduced by analysing its bricks and braces. (For instance, 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 , we adopt the notation to denote its color classes. We will generally use letters and to denote the color classes; sometimes we may instead use and . As shown in Figure 3, members of (or of ) will be denoted using letters and (with subscripts and/or superscripts) and will be depicted using hollow nodes; likewise, members of (or of ) will be denoted using letters and (with subscripts and/or superscripts) and will be depicted using solid nodes.
The neighborhood of a set of vertices is denoted by . The following may be deduced from the well-known Hall’s Theorem.
Proposition 1.1
For a connected bipartite graph , where , the following are equivalent:
- (i)
* is matching covered,* 2. (ii)
* for every nonempty proper subset of , and* 3. (iii)
* has a perfect matching for each pair of vertices and . *
Suppose that is an odd subset of the vertex set of a connected bipartite graph . Then one of the two sets and is larger than the other; the larger set, denoted , is called the majority part of ; the smaller set, denoted , is called the minority part of . 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 of a bipartite matching covered graph is tight if and only if the following hold:
- (i)
* is odd and ; consequently ; and* 2. (ii)
*there are no edges between and . *
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 of order six or more, where , the following are equivalent:
- (i)
* is a brace,* 2. (ii)
* for every nonempty subset of such that ,* 3. (iii)
* has a perfect matching for any four distinct vertices and ,* 4. (iv)
* is -extendable. *
Thus braces (of order at least four) are precisely those bipartite graphs that are -extendable. The following immediate consequence of Propositions 1.1 and 1.3 is worth noting.
Corollary 1.4
Let denote a bipartite graph obtained from a graph by adding an edge. If is matching covered then so is . Furthermore, if is a brace then so is .
The braces and are the only simple bipartite matching covered graphs of order at most four. For a bipartite matching covered graph of order six or more, it is easy to show that if has a -vertex-cut then has a nontrivial tight cut. This implies the following.
Proposition 1.5
Every brace, of order six or more, is -connected.
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 (where ), denoted , is the simple bipartite graph obtained from the cycle graph by adding two nonadjacent vertices — each of which has degree exactly . Observe that has size . The cube , shown in Figure 3(a), is the smallest biwheel. Except for , 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 is the smallest prism. The smallest Möbius ladders are and the brace 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 of a matching covered graph is removable if 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.
Now let denote a brace of order six or more, and let . The bipartite matching covered graph may not be a brace. In particular, one or both ends of may have degree precisely two (in ); 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 be a matching covered graph, and let denote a vertex of degree two that has two distinct neighbors, say and . The bicontraction of is the operation of contracting the two edges and incident with . Note that , where , is a tight cut of . The graph obtained by bicontracting is the same as 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 , denoted , is the matching covered graph obtained by bicontracting all its vertices of degree two that have two distinct neighbors.
For a brace of order six or more, a (removable) edge is thin if 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 is a thin edge of a simple brace , the brace may not be simple. A thin edge of a simple brace is strictly thin if the brace is also simple. For instance, every edge of , and of , 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 denote the set that comprises , and all McCuaig braces. McCuaig’s Theorem [7] may now be stated as follows.
Theorem 1.8
[McCuaig’s Theorem]*
Every simple brace 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 of a simple brace , the index of , denoted , is the number of vertices of degree two in . Clearly, , depending on how many ends of are cubic in . The following is easily verified; see Figures 6 and 7.
Proposition 1.9
Let denote a strictly thin edge of a simple brace , and let . Then and .
1.3 Minimal Braces
Recall that a brace is minimal if, for each , the graph is not a brace. An edge of a simple brace is superfluous if 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 denote a brace of order six or more. If the set of all noncubic vertices is a stable set then is a minimal brace.
The graph shown in Figure 3(b), obtained from 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 shown in Figure 1(b), obtained from 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 denote a minimal brace that is not a member of . By McCuaig’s Theorem, has a strictly thin edge, say . We let denote the simple brace . However, may not be a minimal brace. Clearly, we may choose a set such that is a minimal brace. (In this manner, we may recover a smaller minimal brace .) Note that , each member of is a superfluous edge of , and that if and only if is a minimal brace. This brings us to the following definition.
Definition 1.11
[Minimality-Preserving Pair]* For a minimal brace , a pair is a minimality-preserving pair if is a strictly thin edge of , and is a subset of so that the graph is a minimal brace.*
In the above definition, since each edge of naturally corresponds to an edge of , one may instead view the set as a subset of . Figure 3 shows an example of a minimal brace with a minimality-preserving pair where .
The following is a trivial consequence of McCuaig’s Theorem.
Corollary 1.12
Every minimal brace has a minimality-preserving pair for any strictly thin edge .
The next statement follows immediately from Proposition 1.9.
Proposition 1.13
Let denote a minimality-preserving pair of a minimal brace , and let . Then and .
Corollary 1.12 may be viewed as an induction tool for minimal braces; however, it is not particularly useful for the following reason. If is a minimality-preserving pair of a minimal brace , then the minimal brace can be arbitrarily smaller in size than depending on the cardinality of the set . (This is in contrast to McCuaig’s Theorem; see Proposition 1.9). On the other hand, it seems intuitive that for a minimal brace one should be able to find a minimality-preserving pair such that the set is “small”. This is in fact a consequence of our Main Theorem (3.12).
Corollary 1.14
Every minimal brace has a minimality-preserving pair such that .
Apart from this quantitative information regarding the minimality-preserving pair , our Main Theorem (3.12) also provides qualitative information: for instance, each member of is at distance one from the strictly thin edge .
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 for any minimal brace , where , 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 denote any (removable) edge of a simple brace of order six or more. Note that is not superfluous if and only if the bipartite matching covered graph has a nontrivial tight cut. One may now easily deduce the following from Proposition 1.2.
Corollary 2.1
Let denote a simple brace of order six or more. An edge of is not superfluous if and only if there exist partitions of and of such that and is the only edge joining a vertex in to a vertex in .
The brace , shown in Figure 1(b), has precisely one edge joining two noncubic vertices, and it is not superfluous; consequently, is a minimal brace.
For a simple bipartite graph , its bipartite complement is the graph that has the same set of vertices and has edge set . Using this notion, one may easily prove that and 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, is the only simple brace of order six. Using Corollary 2.1, one may infer that is the only minimal brace of order eight.
Proposition 2.2
The only minimal braces of order at most eight are and .
Proposition 2.3
The only simple cubic bipartite graphs of order ten are and .
Proof: Let denote a simple cubic bipartite graph of order ten, and let denote its bipartite complement. Clearly, is -regular. Observe the following. If is disconnected then is isomorphic to . Otherwise is connected and is isomorphic to .
Proposition 2.4
The only minimal braces of order ten are , and .
Proof: Let denote a minimal brace of order ten. If is cubic then, by Proposition 2.3, is isomorphic to . Now assume that is noncubic, and let denote the set of noncubic vertices.
First suppose that is a stable set. Observe that has precisely two members, say and , each of which has degree precisely four. Consequently, is a connected -regular bipartite graph. (Recall that braces of order six or more are -connected.) Thus is isomorphic to and is isomorphic to .
Now suppose that is not a stable set, and let denote an edge that joins and . Since is devoid of superfluous edges, the edge in particular is not superfluous. By Corollary 2.1, there exist partitions of and of such that and is the only edge joining a vertex in to a vertex in . Since and are noncubic vertices, we infer that each of the sets and has at least three vertices. Consequently, , and each of the induced subgraphs and is isomorphic to . Since , by Proposition 1.3, the induced subgraph has a perfect matching. All of these facts imply that the brace is a subgraph of , whence is isomorphic to .
Definition 2.5
[Stable-Extension]* Let denote a stable set of a connected bipartite graph that meets each color class in precisely two vertices. We let where and . The graph obtained from — by adding two new vertices and , and five new edges and — is called the stable-extension of with respect to — or simply the -extension of . We refer to and as the extension vertices of .*
The graph , shown in Figure 4(a), has a unique stable set that meets each color class in precisely two vertices. We let , shown in Figure 4(b), denote the -extension of . Using Propositions 1.3 and 1.10, one may verify that is a minimal brace.
Proposition 2.6
The only minimal braces of order , and size at least , are and .
Proof: Let denote a minimal brace of order and size at least , and let denote the set of noncubic vertices. Clearly, each of the sets and is nonempty. We let denote the set of edges that have both ends in .
First suppose that is a stable set (i.e., ), whence has maximum degree at most five. Observe that if there exists a vertex of degree five then ; whence is a connected -regular bipartite graph; consequently, is isomorphic to and is isomorphic to . Otherwise, , and each member of has degree precisely four. In this case, observe that is a -regular (bipartite) graph; thus, has four components (each isomorphic to ), and is isomorphic to .
Now suppose that is not a stable set. Our goal is to arrive at a contradiction; however, it requires some tedious arguments.
Let denote any member of . Since is not superfluous, by Corollary 2.1, there exist partitions of and of such that and is the only edge joining a vertex in to a vertex in . Since each end of is noncubic, and since , one of the two sets and has cardinality three and the other one has cardinality four. Adjust notation so that and . See Figure 5(a). Let denote the unique member of . Observe that and .
In particular, we have proved the following.
2.6.1
Each edge has an end whose degree is precisely four, say , such that there exists a cubic vertex that satisfies .
The degree of vertex is either four or five; we will prove that it must be four. Suppose to the contrary that has degree five; whence . By a simple counting argument, there exists a noncubic vertex, say , in . Since is a member of , using statement 2.6.1, we infer that has degree precisely four, and there exists a cubic vertex that satisfies . Since each member of is adjacent with , it follows that . Consequently, . This implies that . This contradicts Proposition 1.3. Thus has degree four.
In particular, we have now proved the following.
2.6.2
For every edge , each end of has degree precisely four. (Consequently, , and each vertex in has degree precisely four.)
Now, we will prove that each neighbor of , distinct from , is cubic. Suppose to the contrary that there exists that is noncubic. Thus has degree precisely four and is a member of . Now we invoke statement 2.6.1. Either there exists a cubic vertex that satisfies , or otherwise there exists a cubic vertex that satisfies . In the latter case, note that (since each vertex in is adjacent with ); however, this implies that we have an edge joining and ; contradiction. In the former case, note that ; whence , and this contradicts Proposition 1.3. Thus each neighbor of , distinct from , is cubic; see Figure 5(b).
In particular, we have established the following.
2.6.3
For every edge , each end of has degree precisely four (in ) and at least one of them has degree precisely one in the induced subgraph .
We let and denote two vertices of that are distinct from and . Note that and . By Proposition 1.3, the graph has a perfect matching; whence the three edges joining and constitute a matching. This implies that has precisely one neighbor in . Consequently, . Observe that the induced subgraph contains a path of length three: . The edge contradicts statement 2.6.3.
This completes the proof of Proposition 2.6.
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 , and nonadjacent vertices and , denotes the graph obtained from by adding the edge . Note that if is a brace then, by Corollary 1.4, is a (simple) brace and is a strictly thin edge of index zero; in this case, we say that is obtained from 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 denote a noncubic vertex of a simple bipartite matching covered graph . Adjust notation so that . Suppose that a (bipartite) graph is obtained from by replacing the vertex by two new vertices and , distributing the edges in incident with between and such that each gets at least two edges, and then adding a new vertex and two new edges: . We say that is obtained from by bi-splitting into , and we denote this as . It is easily verified that is also matching covered. Observe that can be recovered from by bicontracting the vertex and denoting the contraction vertex by .
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 and , of a simple brace that belong to the same color class such that at least one of them, say , is noncubic. A graph is obtained from by an expansion of index one if .*
As an example, one may construct the brace from the brace 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 as per the labeling in Figure 4(a). Now, observe that, as per the labeling in Figure 4(b), .333In order to be consistent with the labeling used in Figures 4(a) and 4(b), we slightly abuse notation by overloading the label .
Definition 3.2
[Expansion of Index Two]* Choose two noncubic vertices, say and , of a simple brace that lie in distinct color classes. A graph is obtained from by an expansion of index two if .*
For convenience, we say that a graph is obtained from the simple brace by an expansion operation if is obtained from by an expansion of index zero, one or two. McCuaig [7] proved the following.
Proposition 3.3
Any graph , that is obtained from a simple brace by an expansion operation, is also a simple brace.
Let denote a simple brace. Observe the following. If for some , then is a strictly thin edge of index one. On the other hand, if for some and , then is a strictly thin edge of index two. In either case, is isomorphic to . We now state the ‘generation version’ of McCuaig’s Theorem (that is equivalent to Theorem 1.8.)
Theorem 3.4
Every simple brace 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 where is a strictly thin edge of a simple brace . In , 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 has precisely inner vertices and outer vertices; see Figures 6 and 7.
3.2 Index one
Throughout this subsection, we assume that is a minimal brace, and that is a strictly thin edge of index one where is the cubic end of . We also assume that the simple brace is not minimal; whence has superfluous edge(s). Thus and . We let and denote the outer vertices of . See Figure 6.
Lemma 3.5
Let denote any superfluous edge of . Then has a cubic outer vertex that is an end of , whereas the other end of is noncubic.
Proof: Recall definition 3.1, and observe the following. If each of and has degree at least three in , then may be obtained from the brace by an expansion of index one; whence is a brace (by Proposition 3.3); contradiction. It follows that one of and is a cubic end of . Clearly, the other end of is noncubic.
Lemma 3.6
Let and denote two edges of such that is a brace. Then has a cubic outer vertex that is a common end of and .
Proof: It follows from the hypothesis that each of and is a superfluous edge of . By Lemma 3.5, for , the edge has precisely one cubic end (in ) that lies in . If the cubic end of is the same as the cubic end of , then there is nothing to prove.
Now suppose that for . Consequently, and are both cubic (in ), whence has degree precisely four in ; consequently, is a vertex of degree two in ; contradiction.
Note that, if is a minimality-preserving pair of then any two distinct members of , say and , satisfy the hypothesis of Lemma 3.6. We thus have the following immediate consequence.
Corollary 3.7
If is a minimality-preserving pair of , then has a cubic outer vertex such that . Consequently, .
Lemma 3.8
Suppose that is a minimality-preserving pair of such that . Then, for each , the noncubic end of is not adjacent with the noncubic end of . Moreover, is isomorphic to a stable-extension of the minimal brace .
Proof: Let . We invoke Lemma 3.6, and adjust notation so that is cubic and . We let and denote the noncubic ends of and , respectively. We let . Thus is a minimal brace. See Figure 8.
Our goal is to prove that . By symmetry, it suffices to prove that . Suppose to the contrary that . See Figure 9.
In this case, and are adjacent in as well. Observe that one may obtain the graph from the brace by an expansion of index one. In particular, . By Proposition 3.3, is a brace — contrary to our hypothesis that is minimal.
We thus conclude that . Consequently, is a stable set of that meets each color class in two vertices. Observe that is isomorphic to the -extension of (with and playing the role of the extension vertices). This completes the proof of Lemma 3.8.
3.3 Index two
Throughout this subsection, we assume that is a minimal brace that is devoid of strictly thin edges of index one, and that is a strictly thin edge of index two. We also assume that the simple brace is not minimal; whence has superfluous edge(s). Thus and . We let and denote the outer vertices of . See Figure 7.
Since is a strictly thin edge, has at most one edge that has one end in and the other end in ; furthermore, has (precisely) one such edge if and only if .
Lemma 3.9
Let denote any superfluous edge of . Then has a cubic outer vertex that is an end of ; the other end of is either noncubic, or it is another cubic outer vertex.
Proof: Recall definition 3.2, and observe the following. If each of has degree at least three in , then may be obtained from the brace by an expansion of index two; whence is a brace (by Proposition 3.3); contradiction. It follows that one of is a cubic end of . Adjust notation so that is a cubic end of . Clearly, if the other end of is not in then it is noncubic.
Now suppose that the other end of is in ; adjust notation so that . Assume that is noncubic. Let denote the neighbor of that is distinct from and . See Figure 10. Observe that the graph may be obtained from the brace by an expansion of index one; in particular, . Thus is a brace (by Proposition 3.3). Consequently, is a strictly thin edge of index one, contrary to our hypothesis that is devoid of such edges. Thus is also cubic.
Lemma 3.10
Let denote any superfluous edge of . Then, in , either joins two outer vertices, or otherwise is adjacent with an edge that joins two outer vertices.
Proof: By Lemma 3.9, has a cubic outer vertex that is an end of . Adjust notation so that is a cubic end of . If the other end of lies in then we are done.
Now suppose that , and let denote the noncubic end of . Assume that . Let denote the neighbor of that is distinct from and . Note that . See Figure 11. Observe that the graph may be obtained from the brace by an expansion of index one; in particular, . Thus is a brace (by Proposition 3.3). Consequently, is a strictly thin edge of index one, contrary to our hypothesis that is devoid of such edges. Thus, one of and is an edge of ; whence is adjacent with an edge that joins two outer vertices.
Corollary 3.11
If is a minimality-preserving pair of , then has two adjacent outer vertices and such that . Furthermore, if then is cubic; likewise, if then is cubic. Consequently, .
3.4 Main Theorem
We are now ready to state and prove the Main Theorem.
Theorem 3.12
[Main Theorem]* Every minimal brace has a minimality-preserving pair such that either , or otherwise satisfies the following properties in :*
- (i)
If then has an outer vertex, say , such that . 2. (ii)
If then has two adjacent outer vertices, say and , such that . 3. (iii)
For each , an end of is cubic if and only if it is an outer vertex.
Consequently, . Furthermore, if and then is isomorphic to a stable-extension of the minimal brace .
Proof: By McCuaig’s Theorem and its corollary (1.12), has a strictly thin edge (of index one or two); furthermore, for any such edge , there exists a minimality-preserving pair .
If has a strictly thin edge of index one, say , then we choose any minimality-preserving pair , and we are done by invoking Lemma 3.5, Corollary 3.7 and Lemma 3.8.
Now suppose that is devoid of strictly thin edges of index one. Thus has a strictly thin edge of index two, say . We choose any minimality-preserving pair , and we are done by invoking Lemma 3.9 and Corollary 3.11.
For convenience, let us introduce the following definition.
Definition 3.13
[Narrow Minimality-Preserving Pair]* A minimality-preserving pair of a minimal brace 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 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 and the index of the strictly thin edge ). 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 is isomorphic to a stable-extension of a simple brace , then may be obtained from 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 , that is isomorphic to a stable-extension of a simple brace , is also a simple brace.
4 An application
In this section, as an application of the Main Theorem (3.12), we will prove that if is a minimal brace (where ) then ; furthermore, we shall provide a complete characterization of minimal braces that meet this upper bound.
We begin by defining an infinite family of minimal braces, each of whose members meets this upper bound; its smallest member is . For , we define the graph as the -extension of — where the set comprises the noncubic vertices of . Now let . Since is a simple brace, it follows from Corollary 3.14 that each member of is a simple brace. Furthermore, by Proposition 1.10, we infer that each member of is in fact a minimal brace. By summing degrees, observe that if then .
We now prove the following easy corollary of the Main Theorem.
Corollary 4.1
Let denote a narrow minimality-preserving pair of a minimal brace , and let . Assume that . Then the following hold:
- (i)
. 2. (ii)
If then , and ; consequently, is isomorphic to a stable-extension of .
Proof: By Proposition 1.13, and . By Theorem 3.12, . We now consider two cases depending on the index of .
First consider the case: . Using the equations and inequalities noted above, we have: Thus, in this case, the strict inequality holds.
Now consider the case: . Using the same equations and inequalities as before, we have: . Hence, we have the inequality ; equality holds if and only if and . By the last part of Theorem 3.12, we infer that is isomorphic to a stable-extension of . This completes the proof of Corollary 4.1.
We are now ready to prove the main result of this section.
Theorem 4.2
Let denote a minimal brace that is not in . Then , and equality holds if and only if .
Proof: We proceed by induction on the order of the graph. Let denote a minimal brace that is not in .
The reader may verify, using Propositions 2.2 and 2.4, that the desired conclusion holds when . Now suppose that . If is cubic then . We may thus assume that is noncubic. Now one may verify, using Proposition 2.6, that the desired conclusion holds when . Henceforth, suppose that . If is a biwheel then . We may thus assume that is not a biwheel. Consequently, .
By the Main Theorem (3.12), has a narrow minimality-preserving pair, say . Let denote the minimal brace . Note that where ; whence . By invoking the induction hypothesis, either , or otherwise and equality holds if and only if .
First suppose that . In particular, and . Consequently, and . By Proposition 1.13, where . It follows that . The desired conclusion holds.
Now suppose that and equality holds if and only if . By invoking Corollary 4.1, we infer that . Since , it only remains to prove the following: if then .
Henceforth, assume that . It now follows from Corollary 4.1 that . Thus . Furthermore, has a stable set that meets each color class in precisely two vertices, and is isomorphic to the -extension of . We let and denote the color classes of , and we let and denote the extension vertices of . Thus and .
One may easily verify that , shown in Figure 2(a), has no stable set that meets each color class in two vertices. Thus, either is isomorphic to or otherwise .
First consider the case in which is isomorphic to . We let and denote the noncubic vertices of . The reader may easily verify that, up to symmetry, has only one stable set that meets each color class in precisely two vertices — as shown in Figure 12(a). Thus, we let ; whence is isomorphic to the graph shown in Figure 12(b). We shall now arrive at a contradiction by showing that is a brace — where . By Proposition 1.3, for each nonempty set such that , the inequality holds; furthermore, it suffices to verify that holds; by comparing the graphs and , observe that we only need to check those sets that satisfy the following: . The reader may verify that, for each set that satisfies , the inequality holds. Thus is a superfluous edge in ; contradiction. (In fact, by symmetry, any edge of , both of whose ends are noncubic, is superfluous.)
Now consider the case in which . We let and — such that is the set of noncubic vertices of , and for each — as shown in Figure 13(a). We consider two subcases depending on whether the set intersects with the stable set or not.
First suppose that . Thus, we may adjust notation so that ; see Figure 13(b). Similarly to an earlier case, we will arrive at a contradiction by showing that is a brace — where . By Proposition 1.3, for each nonempty set such that , the inequality holds; furthermore, it suffices to verify that holds; by comparing the graphs and , observe that we only need to check those sets that satisfy the following: . The reader may verify that, for each set that satisfies , the inequality holds. Thus is a superfluous edge in ; contradiction. (In fact, by symmetry, any edge of , both of whose ends are noncubic, is superfluous.)
Now suppose that . Observe that, in this case, . Consequently, , by definition of the family . This completes the proof of Theorem 4.2.
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.
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