
TL;DR
This paper introduces minimal bricks, a special class of 3-connected graphs with unique matching properties, and proves a generation theorem along with bounds on edges and degree conditions.
Contribution
It provides a generation theorem for minimal bricks and establishes bounds on edges and degree properties, advancing understanding of their structure.
Findings
Minimal bricks have at most 5n-7 edges for n>4.
Every minimal brick has at least three vertices of degree three.
A generation theorem for minimal bricks is proved.
Abstract
A brick is a 3-connected graph such that the graph obtained from it by deleting any two distinct vertices has a perfect matching. A brick is minimal if for every edge e the deletion of e results in a graph that is not a brick. We prove a generation theorem for minimal bricks and two corollaries: (1) for n>4, every minimal brick on 2n vertices has at most 5n-7 edges, and (2) every minimal brick has at least three vertices of degree three.
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Minimal bricks
Serguei Norine
and
Robin Thomas
School of Mathematics, Georgia Tech, Atlanta, GA 30332-0160
Abstract.
A brick is a -connected graph such that the graph obtained from it by deleting any two distinct vertices has a perfect matching. A brick is minimal if for every edge the deletion of results in a graph that is not a brick. We prove a generation theorem for minimal bricks and two corollaries: (1) for , every minimal brick on vertices has at most edges, and (2) every minimal brick has at least three vertices of degree three.
16 April 2019. Partially supported by NSF grants 0200595 and 0354742. Published in *J. Combin. Theory Ser. B *96 (2006), 505–513. This version fixes an error kindly pointed to us by P. A. Fabres, N. Kothari and M. H. de Carvalho.
1. Introduction
All the graphs considered in this paper are finite and simple. A brick is a -connected graph such that the graph obtained from it by deleting any two distinct vertices has a perfect matching. The importance of bricks stems from the fact that they are building blocks of the matching decomposition procedure of Kotzig, and Lovász and Plummer [6]. In particular, many matching problems of interest (such as, for example, computing the dimension of the linear hull [3] or lattice [5] of incidence vectors of perfect matchings, or characterizing graphs that admit a “Pfaffian orientation” [8]) can be reduced to bricks.
In an earlier paper we proved a generation theorem for bricks. The precise statement requires a large number of definitions, and is given in Theorem 2.3 below. Let us describe the result informally first. Let be a graph, and let be a vertex of of degree two incident with the edges and . Let be obtained from by contracting both and and deleting all resulting parallel edges. We say that was obtained from by bicontracting or bicontracting the vertex , and write . A subgraph of a graph is central if has a perfect matching. We say that a graph is a matching minor of a graph if can be obtained from a central subgraph of by repeatedly bicontracting vertices of degree two. We denote the fact that is isomorphic to a matching minor of by writing . Our generation theorem of [7] asserts that, except for a few well-described exceptions, if , then a graph isomorphic to can be obtained from by repeatedly applying a certain operation in such a way that all the intermediate graphs are bricks and no parallel edges are produced. The operation is as follows: first delete an edge, and for every vertex of degree two that results contract both edges incident with it. The theorem improves a recent result of de Carvalho, Lucchesi and Murty [2], but in this paper we seem to need our result.
We found our theorem useful for generating interesting examples of bricks and testing various conjectures, but even more useful was a variant for minimal bricks, which we prove in this paper. A brick is minimal if is not a brick for every edge . (We use for deletion.) The theorem asserts that every minimal brick other than the Petersen graph can be obtained from or the prism (the complement of a cycle of length six) by taking “strict extensions” in such a way that all the intermediate graphs are minimal bricks not isomorphic to the Petersen graph. The theorem is formally stated as Theorem 3.2. We postpone the definition of strict extensions until they are needed.
The paper is organized as follows. In the next section we introduce the results from [7] that we need. In Section 3 we state and prove our generation theorem for minimal bricks; we deduce it from the more general Theorem 3.1. In Section 4 we prove that, except for four graphs on at most eight vertices, every minimal brick on vertices has at most edges. Finally, in Section 5 we prove that every minimal brick has at least three vertices of degree three.
2. The tools
In this section we state the results of [7] that we need, but let us start with the following theorem of Lovász [4]; see also [6, Theorem 5.4.11].
Theorem 2.1**.**
Every brick has a matching minor isomorphic to or the prism.
The theorem of de Carvalho, Lucchesi and Murty [2] mentioned in the introduction uses and the prism as the starting graphs of their generation procedure. We use a more restricted set of operations, and the price we pay for that is that the starting set has to be expanded. We now introduce the relevant classes of graphs.
Let and be two vertex-disjoint cycles of length with vertex-sets ,, and , (in order), respectively, and let be the graph obtained from the union of and by adding an edge joining and for each . We say that is a planar ladder. Let be the graph consisting of a cycle with vertex-set ,, (in order), where is an integer, and edges with ends and for ,. We say that is a Möbius ladder. A ladder is a planar ladder or a Möbius ladder. Let be a planar ladder as above on at least six vertices, and let be obtained from by deleting the edge and contracting the edges and . We say that is a staircase. Let be an integer, and let be a path with vertices in order. Let be obtained from by adding two distinct vertices and edges and for and all even and and all odd . Let be obtained from by adding the edge . We say that is an upper prismoid, and if , then we say that is a lower prismoid. A prismoid is a lower prismoid or an upper prismoid.
We need the following strengthening of Theorem 2.1, proved in [7, Theorem (1.8)].
Theorem 2.2**.**
Let be a brick not isomorphic to , the prism or the Petersen graph. Then has a matching minor isomorphic to one of the following seven graphs: the graph obtained from the prism by adding an edge, the lower prismoid on eight vertices, the staircase on eight vertices, the staircase on ten vertices, the planar ladder on ten vertices, the wheel on six vertices, and the Möbius ladder on eight vertices.
In the introduction we described our generation theorem by means of operations that reduce the larger graph to its matching minor . This version is easier to describe concisely, but for both the proof and the applications it is better to proceed the other way, namely to describe how to obtain from . Thus we reverse the process now and proceed in the other direction. Here are the relevant definitions.
Let be as in the definition of bicontraction. Assume that , are not adjacent, that they both have degree at least three and that they have no common neighbors except ; then no parallel edges are produced during the contraction of and . Let be the new vertex that resulted from the contraction. We say that was obtained from by bisplitting the vertex . We call the new inner vertex and and the new outer vertices. Let be a graph. We wish to define a new graph and two vertices of . Either and are two nonadjacent vertices of , or is obtained from by bisplitting a vertex, is the new inner vertex of and is not adjacent to , or is obtained by bisplitting a vertex of a graph obtained from by bisplitting a vertex, and and are the two new inner vertices of . Finally, let be obtained from by adding an edge with ends . We say that is a linear extension of .
Since in the next theorem the graph need not be a brick we need two more exceptional classes of graphs. Let be an even cycle with vertex-set in order, where is an integer and let be obtained from by adding vertices and and edges joining to the vertices of with odd indices and to the vertices of with even indices. Let be obtained from by adding an edge . We say that is an upper biwheel, and if we say that is a lower biwheel. A biwheel is a lower biwheel or an upper biwheel. Please note that biwheels are bipartite, and therefore are not bricks.
We are now ready to state a version of our generation theorem [7, Theorem (1.10)]. The version mentioned in the introduction follows easily, because a linear extension of a brick is a brick.
Theorem 2.3**.**
Let be a brick other than the Petersen graph, and let be a -connected matching minor of . Assume that if is a planar ladder, then there is no strictly larger planar ladder with , and similarly for Möbius ladders, wheels, lower biwheels, upper biwheels, staircases, lower prismoids and upper prismoids. If is not isomorphic to , then some matching minor of is isomorphic to a linear extension of .
3. Generation Theorem for Minimal Bricks
In this section we prove a generation theorem for minimal bricks, Theorem 3.2 below. We derive it from the more general Theorem 3.1.
If is a graph, and are distinct nonadjacent vertices, then or denotes the graph obtained from by adding an edge with ends and . If and are adjacent or equal then . Now let be adjacent. By bisubdividing the edge we mean replacing the edge by a path of length three, say a path with vertices , in order. Let be obtained from by this operation. We say that (in that order) are the new vertices. Thus are the new vertices resulting from subdividing the edge (we are conveniently exploiting the notational asymmetry for edges). Now if , then by we mean the graph . Notice that the graphs and are different.
Let be a graph, let be distinct, and let be obtained from by bisubdividing , where the new vertices are . Let and be not necessarily distinct vertices such that not both belong to . In those circumstances we say that is a quasiquadratic extension of . We say that it is a quadratic extension of if and are not adjacent in . (Recall our convention that if and are adjacent in , then .) We say that is the base of this quasiquadratic extension.
Now let be as above, and let be not necessarily distinct vertices such that , and if then . If , then let be obtained from by bisubdividing , and let be the new vertices. If , then let be obtained from by adding new vertices and edges , and . Then the graph is called a quasiquartic extension of . It is a quartic extension of if . We say that are the bases of the quasiquartic extension. Quadratic and quartic extensions were used in the proof of Theorem 2.3 in [7]; quasiquadratic and quasiquartic extensions are new.
We need to define two new types of extension. We say that a linear extension of a graph is strict if . Let be pairwise distinct vertices of , let be obtained from by bisplitting , and let be the new inner vertex and a new outer vertex. If and then the graph , where are the new vertices of , is called a bilinear extension of . If then the graph , where are the new vertices of , is called a pseudolinear extension of . See Figure 1.
Finally, we say that is a strict extension of if is a quasiquadratic, quasiquartic, bilinear, pseudolinear or strict linear extension of . It is not hard to see that a strict extension of a brick is a brick.
Theorem 3.1**.**
Let be a brick other than the Petersen graph, and let be a -connected matching minor of such that . Then some matching minor of is isomorphic to a strict extension of .
**Proof. **Let a graph be chosen so that is a spanning subgraph of , and is maximal.
Suppose first that is a planar ladder and there exists a planar ladder with and . Then clearly , and if we choose with minimum, then is a quartic extension of and therefore the theorem holds. Therefore we can assume that if is a planar ladder, then there is no strictly larger planar ladder with , and similarly for Möbius ladders, wheels, lower biwheels, upper biwheels, staircases, lower prismoids and upper prismoids. By Theorem 2.3 and the choice of there exists a strict linear extension of such that . We denote by and break the analysis into cases depending on the type of strict linear extension.
Suppose first that , where is obtained from by bisplitting a vertex, is the new inner vertex of and . Let and be the new outer vertices. We have , in the natural way. For let be the number of edges of that are incident with in (or ). We assume without loss of generality that . Note that , because has degree at least three in .
If then is a strict linear extension of . If let be an edge incident with ; then is a quadratic extension of . Finally, if and are incident with then is a quasiquadratic extension of .
Now suppose , where is obtained by bisplitting a vertex of a graph obtained from by bisplitting a vertex, and and are the two new inner vertices of . Let and , respectively, be the corresponding new outer vertices. Let and be defined analogously as above. We start by assuming that and are pairwise distinct and without loss of generality assume , .
If then is a strict linear extension of . If then is isomorphic to a strict linear extension of unless the edge of incident with is incident also with one of the vertices and . In this case is a bilinear extension of , for every incident with . If for let denote the unique edge in incident with and let denote some edge in incident with . If then is a quasiquartic extension of . (If is adjacent to , then we need the provision of in the definition of quasiquartic extension.) Otherwise, without loss of generality we assume that is not incident with and deduce that is a quadratic extension of with base .
It remains to consider the subcase when . Let be incident with such that has no end in . If then is a strict linear extension of . If let denote the unique edge in incident with . If is not incident with then is a quasiquadratic extension of if is not incident with and is a quasiquartic extension of if is incident with . If on the other hand is incident with then is a quadratic extension of , where is any edge in incident with . Finally, if let be incident with and have no end in . Then is a quasiquadratic extension of . This completes the case when and are pairwise distinct.
We now assume without loss of generality that . Then and are pairwise distinct and we assume , again without loss of generality. Suppose first . If then is a pseudolinear extension of , where is incident with ; if then is a quadratic extension of and if then is a quasiquadratic extension of , where is an edge in incident with and not adjacent to . Therefore we may assume . If and then or is a strict linear extension of . If and then is a quadratic extension of , where is as above. If, finally, then let be obtained from by deleting edges of incident with and edges incident with ; in that case is a quasiquadratic extension of .
This completes the case analysis. ∎
Theorem 3.1 implies the following generation theorem for minimal bricks.
Theorem 3.2**.**
Let be a minimal brick other than the Petersen graph. Then can be obtained from or the prism by taking strict extensions, in such a way that all the intermediate graphs are minimal bricks not isomorphic to the Petersen graph.
**Proof. **Suppose the statement of the theorem is false and let be a counterexample with minimum.
By Theorem 2.1 we may choose a minimal brick such that can be obtained from or the prism by taking strict extensions and, subject to that, is maximum. If then is isomorphic to by the minimality of . If, on the other hand, , then by Theorem 3.1 there exists a strict extension of . Let be a minimal brick with ; then . It follows that is not isomorphic to , for otherwise so is , contrary to our assumption that is a counterexample to the theorem. By the minimality of the graph can be obtained from or the prism by taking strict extensions, contrary to the choice of . ∎
Note that there exist bricks obtained from or the prism by a sequence of strict extensions, that are not minimal. A simple example follows.
Let be the prism, , the vertices ,, are pairwise adjacent and so are the vertices , and is adjacent to for . Let and let , where are the new vertices of . Then is a quasiquadratic extension of and is a brick, which can be obtained from a prism by a quadratic extension or a sequence of two linear extensions.
4. Edge Bound for Minimal Bricks
The following theorem is [6, Corollary 5.4.16].
Theorem 4.1**.**
If is a minimal bicritical graph with vertices, then .
We use Theorem 3.1 to prove a similar bound for minimal bricks.
Theorem 4.2**.**
Let be a minimal brick on vertices. Then , unless is the prism or the wheel on four, six or eight vertices.
**Proof. **The theorem holds for the Petersen graph, so from now on we assume that is not the Petersen graph, the prism or the wheel on six or eight vertices. Denote the last three graphs by , and , respectively.
Note that a strict linear extension increases the number of vertices in a graph by or and the number of edges by or , respectively. Similarly, a quasiquadratic extension increases the number of vertices by and the number of edges by at most , while quasiquartic, bilinear and pseudolinear extensions increase the number of vertices by and the number of edges by at most .
We say that a brick is sparse if and we say that is dense otherwise. We claim that any minimal brick that contains a sparse matching minor is sparse. Suppose and are bricks, , is sparse and is minimal. Let a sparse brick be chosen with maximum. From Theorem 3.1 we deduce that either or some strict extension of is a matching minor of . In the latter case, by the calculations above, is sparse in contradiction with the choice of . Therefore and is isomorphic to by the minimality of . The claim follows.
Suppose is dense. By Theorem 2.2 has a matching minor isomorphic to one of the seven graph mentioned therein, and hence has a matching minor isomorphic to one of the following four graphs: , , the staircase on eight vertices, and the Möbius ladder on eight vertices. Among these graphs only two are dense: and .
Assume first that contains as a matching minor. By Theorem 3.1 there exists a strict extension of the prism such that . By the calculations above is sparse, unless is a quadratic extension of with base , where . We will show that there exists such that is a brick. Note that is sparse. Therefore it follows that any minimal brick containing the prism as a matching minor and not equal to it is sparse. We prove the existence of by listing all possible quasiquadratic extensions of with edges in Figure 2. An edge that satisfies the conditions above is indicated by a cross. A spanning bisubdivision or bisplit of or in is indicated by bold lines and allows the reader to easily verify that the claim holds in each of the cases.
Therefore we may assume that contains as a matching minor and does not contain . By Theorem 2.3 is a wheel or contains a linear extension of a wheel as a matching minor. All the wheels on at least ten vertices and all strict linear extensions of and are sparse and therefore must contain a graph obtained from or by an edge addition. Every graph obtained from by adding an edge has a matching minor isomorphic to a graph obtained from by adding an edge. The latter graph is unique up to isomorphism and contains as a spanning subgraph, in contradiction with our assumptions. ∎
The bound given in Theorem 4.2 is tight for every . An example of a minimal brick on vertices with edges for follows. Let . For every let ,, and be the edges of . Then for every the graph contains a vertex of degree two, and hence is not a brick. It remains to show that is a brick for every . Note that is a quasiquadratic extension of for every . Therefore it suffices to show that is a brick. The graph is isomorphic to the prism with one of its edges bisubdivided and consequently can be obtained from the prism by a quadratic extension.
5. Three Cubic Vertices
De Carvalho, Lucchesi and Murty [2] proved that every minimal brick has a vertex of degree three. According to them (private communication) it had been conjectured by Lovász. We prove the following strengthening.
Theorem 5.1**.**
Every minimal brick has at least three vertices of degree three.
**Proof. **Let a minimal brick that has at most two vertices of degree three be chosen with minimal. By Theorem 3.2 there exists a minimal brick with at least three vertices of degree three, such that is isomorphic to a strict extension of .
Note that if a strict linear extension is used to obtain from then the degree of at most one vertex of increased and at least one vertex in has degree three. If a quasiquartic, bilinear or pseudolinear extension is used to obtain then contains at least three vertices of degree three. Therefore is isomorphic to a quasiquadratic extension of that is not quadratic.
We assume without loss of generality that and there exist such that , , at least three of the vertices are distinct, and . Note that the vertices of degree three in must form a subset of and that , for the deletion of such an edge from results in a quadratic extension of , contrary to the fact that is a minimal brick.
Since is a brick, it is not a biwheel. By Theorem 2.3 either is a ladder, wheel, staircase or prismoid or is a linear extension of a brick. If is a ladder, wheel, staircase or prismoid distinct from then has at least vertices of degree three, and consequently has at least three vertices of degree three. If then is not minimal, by an observation in the previous paragraph.
Therefore, is a linear extension of a brick, and hence there exists such that becomes a brick after possible bicontractions of vertices of degree two in such a way that no parallel edges are created by these bicontractions. Note that is minimal and therefore at least one end of is a vertex of degree three in . Assume first that exactly one end of has degree three in . Without loss of generality this end is . The graph is a brick, because it can be obtained by a linear extension (first bisplit to produce , then add the edge ) followed by a quadratic extension with base , a contradiction. Recall that is not adjacent to in .
It remains to consider the case when both of the ends of have degree three in . Without loss of generality we assume that , and hence and are pairwise distinct. It follows that is a strict linear extension of and is again a brick. This completes the case analysis. ∎
We conjecture the following strengthening of Theorem 5.1.
Conjecture 5.2**.**
There exists such that every minimal brick has at least vertices of degree three.
Even a much weaker strengthening, namely, the conjecture that every brick has at least four vertices of degree three, seems to require new ideas or a substantial refinement of our techniques.
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