Ear‐decompositions, minimally connected matroids and rigid graphs

We prove that if the two‐dimensional rigidity matroid of a graph G $G$ on at least seven vertices is connected, and G $G$ is minimal with respect to this property, then G $G$ has at most 3 n − 9 $3n-9$ edges. This bound, which is best possible, extends Dirac's bound on the size of minimally 2‐connected graphs to dimension two. The bound also sharpens the general upper bound of Murty for the size of minimally connected matroids in the case when the matroid is a rigidity matroid of a graph. Our proofs rely on ear‐decompositions of connected matroids and on a new lower bound on the size of the largest circuit in a connected rigidity matroid, which may be of independent interest. We use these results to determine the tight upper bound on the number of edges in a minimally redundantly rigid graph in two dimensions. Furthermore, as an application of our proof methods, we give a new proof for Murty's theorem.

A well-known result in extremal graph theory, due to Dirac [3], states that a minimally 2-connected graph on n 4 ≥ vertices has at most n 2 − 4 edges.This bound is the best possible, as shown by the complete bipartite graph K n 2, −2 .Murty [15] proved the following matroidal extension of this result.
Theorem 1.1 (Murty [15,Theorem 3.2]).Let  be a minimally connected matroid on ground-set E with rank function r.Suppose that r E ( ) 3 ≥ .Then , and (ii) if equality holds then  is the graphic matroid of K n 2, −2 , for In this paper, which was motivated by recent results and open questions in rigidity theory, we prove refinements of this theorem for specific matroids defined on the edge set of a graph.The socalled d-dimensional generic rigidity matroid of a graph G, denoted by G ( ) , is defined on the edge set of G.This matroid encodes several fundamental rigidity properties of (generic d-dimensional bar- and-joint realizations of) G.For example, the rigidity of G in d can be characterized by the rank of G ( ) , for all d 1 ≥ .The connectivity (or separability) of this matroid is closely linked to the global rigidity of the graph.For d = 1, 2, a graph G is globally rigid if and only if G is d ( + 1)-connected and G ( ) is connected, see [8].Moreover, the global rigidity of G implies the connectivity of G ( ) , for all d 1 ≥ , see [4].Further connections have been identified in [5], along with a tight upper bound on the number of edges in a minimally globally rigid graph in d .In this context new results concerning graphs with minimally connected rigidity matroids may lead to new structural results on graphs which are minimal with respect to certain (global) rigidity properties.
It is worth noting that G ( )  is equal to the graphic matroid of G, and hence G ( ) 1  is connected if and only if G is 2-connected.Thus our first result can be viewed as the two- dimensional extension of Dirac's theorem.
Theorem 1.2.Let G V E = ( , ) be a graph with   V 7 ≥ , and suppose that G ( ) Thus the bound of Theorem 1.2 is a substantial improvement on that of Theorem 1.1.Furthermore, the graphs K n 3, −3 , for n 7 ≥ , show that the new bound is best possible.
The proof of Theorem 1.2 relies on the fact that every connected matroid has an eardecomposition (which extends the well-known concept of the ear-decomposition of a 2-connected graph).Matroidal ear-decompositions have played a key role in the characterization of globally rigid graphs in 2 in [8] and in some other problems, see [1,19].We shall provide a new proof of Theorem 1.1 by using ear-decompositions.See also [16] for another proof.Our approach using eardecompositions is probably simpler, or at least more convenient than the previous proofs in the context of rigidity matroids as it avoids the operations of taking the dual matroid, or contracting an element, which may result in a matroid that is no longer a rigidity matroid of some graph.
To obtain the tight bound in Theorem 1.2 we need a new structural result on circuits of the twodimensional rigidity matroid, which may be of independent interest.Let K q , 2 | JORDÁN graph on vertex set a b c d i q { , } { , : 1 , see Figure 1.It is easy to check that the two-dimensional rigidity matroids of K q 4 and K ˆq 4 are connected for all q 2 ≥ and their circuits induce at most six vertices.The second main result is as follows.
Then one of the following holds:  -circuit with at least seven vertices, − is rigid for every edge e of G. Redundantly rigid graphs are well-studied objects that occur in several results and applications.In 2 they are sandwiched between rigid graphs and graphs with a connected rigidity matroid.Our third result, whose proof relies on Theorem 1.2, gives a tight bound on the number of edges in a minimally redundantly rigid graph in 2 .
The complete bipartite graphs K n , 7 n 3, −3 ≥ , show that the upper bound is best possible.The higher dimensional versions of our results remain challenging open problems.We shall formulate some questions and conjectures concerning the cases d 3 ≥ , and illustrate the potential difficulties in the application of the ear-decomposition method.

| PRELIMINARIES
Each edge uv of G corresponds to the line segment connecting the points p u ( ) and p v ( ).The framework G p ( , ) is said to be rigid in d if every continuous motion of the vertices that preserves the edge lengths preserves all pairwise distances.It is globally rigid if the edge lengths of G p ( , ) uniquely determine all pairwise distances in d .The framework G p ( , ) is generic if the set of the   d V coordinates of the vertices is algebraically independent over the rationals.It is known that for generic frameworks the (global) rigidity of G p ( , ) in d depends only on G. Thus we may call a graph G rigid (resp., globally rigid) in d if every (or equivalently, if some) generic realization of G in d is rigid (resp., globally rigid).The characterization of (globally) rigid graphs is known for d = 1, 2 and is a major open problem for d 3 ≥ .See [9,10,18] for more details on rigid and globally rigid graphs.
In this paper we shall use a matroidal approach to graph rigidity.For a general introduction to matroid theory we refer the reader to [17].The d-dimensional rigidity matroid of a graph G is a matroid defined on the edge set of G which reflects the rigidity properties of all generic realizations of G in d .See, for example [9] for the definition.We denote the rank of G ( ) The following characterization of rigid graphs is due to Gluck.
A graph is minimally rigid in d if it is rigid in d but deleting any edge results in a graph which is not rigid.Minimally rigid graphs on n vertices in d correspond to the bases of


. Thus they have exactly   ( ) A good combinatorial characterization for the rank function r G ( ) In what follows we shall mostly focus on the two-dimensional case.It will be convenient to define via its set of circuits.Basic results in rigidity theory (see [13]) provide a characterization of independence in G ( ) 2  and imply that an edge set Here H X [ ] denotes the induced subgraph of H on vertex set X .See [9] for more details, including efficient algorithms for testing independence in the two-dimensional rigidity matroid.
Since the rigidity matroid is defined on the edge set of a graph, we can introduce the following graph properties based on this matroid.We say that a graph G Given two graphs H V E = ( , ) 1 1 ∈ , the 2-sum of H 1 and H 2 (along the edge pair

| CONNECTED MATROIDS AND EAR-DECOMPOSITIONS
Let  be a matroid on ground-set E and let be a nonempty sequence of circuits of .
is a partial ear-decomposition of  if for all i t 2 ≤ ≤ the following properties hold: The graph obtained from K 3,5 by adding the edge y y 1 2 .
Ear-decompositions of matroids were introduced in [2], along with some basic properties, such as the first two statements of the following lemma.The third one follows easily from the definitions.
Lemma 3.1.Let  be a matroid with rank function r.Then (a)  is connected if and only if  has an ear-decomposition.(b) If  is connected, then any partial ear-decomposition of  can be extended to an ear- decomposition of .
is an ear-decomposition of , then , that is, the restriction of  to E e − is not connected, for all e E ∈ .It is clear from this definition and Lemma 3.1 that the lobe C ˜t of the last ear of an ear-decomposition of a minimally connected matroid contains at least two elements.As a side remark, note that this observation implies another result of Murty [15,Lemma 3.1], which states that a minimally connected matroid with at least two elements contains a co-circuit of cardinality two: pick two elements from C ˜t.
We can say more: every lobe C ˜i has size at least two, and the restriction of  to the elements of a partial ear-decomposition is minimally connected.Lemma 3.2.Let  be a minimally connected matroid on ground-set be an ear-decomposition of .Then ≥ and e E ∈ .We shall prove that e −  has an ear-decomposition.For each . Thus e −  is connected by Lemma 3.1, which contradicts the minimality of .Thus (i) follows.
To see that (ii) holds suppose that . By Lemma 3.1 there exists an eardecomposition of  D i  for which the lobe of the last ear has only one element (namely, e), and this ear-decomposition can be extended to an ear-decomposition of  by Lemma 3.1.But it is impossible by (i). □ We next give a short proof of Murty's theorem, using ear-decompositions.
By Lemma 3.2 and the minimality of follows, implying that , as required.(ii) To characterize the extremal matroids (i.e., those that satisfy (i) with equality) we first prove three claims concerning their circuits.
Proof.It follows from ( 6) and Lemma 3.1(b) that if   E r E = 2 ( ) − 2 holds then every circuit in  has size at most four.The minimality of  implies that it has no loops or parallel elements.Suppose that C is a circuit in  with   C = 3.Since  is connected and r E ( ) 3 ≥ , there exists a circuit C′ with for some e E ∈ , then we can use the circuit exchange axiom to deduce that there is a circuit ∩ (in which case we are done as above), or a partial ear-decomposition starting with C C ′, * contradicts Lemma 3.2.This proves the claim.
. To see that (a) holds first note that we cannot have   C C = 3 the circuit exchange axiom there is a circuit for some i = 1, 2. But then a partial ear-decomposition starting with C C C , ,

∈
, and a circuit C′ with e f C { , } ′ ⊂ .By using (a) and an argument similar to that of the proof of (a), we can deduce that for some i j q 1 ≤ ≠ ≤ .
be an ear-decomposition of .Since  is extremal, ( 5) and ( 6) show that we must have   C ˜= 2 i for each lobe i t 2 ≤ ≤ , and That is, We prove the existence of the required partition for every , by induction on i.The case i = 1 is obvious, since D C = Suppose that i 3 ≥ .We may assume that i t = .By (7) and induction there is a partition

only if it is the union of two members. Then
□ Note that   E q = 2 and q t = + 1.Now (ii) follows by noting that a matroid satisfying the statement of Claim 3.5 is isomorphic to the graphic matroid of K n 2, −2 for n q (3) give

≤
. These observations lead to the following corollary.and This upper bound is tight for d = 1.However, for d 2 ≥ , the best lower bound we are aware of comes from the complete bipartite graphs

≥
. The fact that ( ) is an d  -circuit and that every edge is incident with a vertex of degree d + 1 implies that these are minimally d  -connected graphs with − ( + 1) 2 edges.We conjecture that this edge count is tight in the following sense.Conjecture 4.2.Let G V E = ( , ) be a minimally d  -connected graph.Then we have where equality holds if and only if G K = d+2 .Furthermore, if   ( ) In the rest of this section and in Section 5 we shall verify Conjecture 4.2 for d = 2 by using ear-decompositions of rigidity matroids.
Let G be an d  -connected graph and consider an ear-decomposition .Then Proof.The proof is by induction on the length t of the ear-decomposition.
1 , and hence follows, as required.
Next suppose that t 2 ≥ .Since the ear-decomposition is strict, we have Y t ≠ ∅.Let G V E ′ = ( ′, ′) be the subgraph of G induced by the union of the edge sets of the first t − 1 ears.The graph G′ is d  -connected and has a strict ear-decomposition , and hence We can show that every ear-decomposition of a minimally 2  -connected graph is strict.The proof is based on the following lemma.
Proof.The proof is based on the fact that 2  -connected graphs are rigid, see, for example [8].This implies that being 2  -connected is a monotone property: adding edges to an 2  -connected graph preserves 2  -connectivity. -circuit of G has exactly four vertices, that is, every 2 implies that each pair of edges in G belongs to a K 4 .Therefore G is a complete graph.Since the only minimally 2  -connected complete graph is K 4 , the theorem follows.□ We shall verify the second part of the conjecture for d = 2 (i.e., the tight upper bound , for   V 7 ≥ ) in Section 5, after proving a new lower bound on the size of the largest 2  -circuit contained by an 2  -connected graph.
We close this subsection with a three-dimensional example which shows that for Lemmas 4.4 and 4.5 do not hold for d 3 ≥ and illustrates the potential difficulties of the ear- decomposition method in higher dimensions.
⊕ (also known as the "double banana" graph), and let G be the graph obtained from B by adding two incident edges e vx f vy = , = for which G e − as well as G f − are rigid, see Figure 3.In G the subgraph B is an 3  -circuit.Another 3  -circuit is the subgraph H of G on six vertices, induced by the vertices of the K 5 that contains x y , , plus v. Let G B ′, ′ and H′ be obtained from G B , and H , respectively, by taking a 2-sum with a copy of K 5 along each edge different from e and f .By using that the 2-sum of two 3  -circuits is an 3  -circuit, it follows that B′ and H′ are 3  -circuits and G′ is 3  -connected.Furthermore, the fact that every edge in E G e f ( ′) − { , } is incident with a vertex of degree four can be used to deduce that G′ is minimally 3  -connected.
It remains an open question whether some ear-decomposition of a minimally d  -connected graph, for d 3 ≥ , is strict.
Recall that the triangular prism, denoted by P, is the complement of the cycle of length six.The addition of any new edge to P results in the same graph, up to isomorphism, that we denote by P e + .The graph obtained from K 3,3 by adding an edge is denoted by K e +  and P e + .
By using the fact that every , and has minimum degree at least three, it is easy to check that there are six different 2  -circuits on at most six vertices, and that they satisfy the following properties.Let K W W K K K e P e = { , , , , + , + }.
⊕ and the two K 4 's share the vertex pair x y , .
We shall use the following lemma.A vertex set X V ⊆ is said to be critical in a graph be the 2  -circuits induced by an ear-decomposition Recall the definition of the graphs K q , 2 q 4 ≥ , on vertex set a b c d i q { , } { , : 1 . We shall say that the vertex pairs c d i q { , }, 1 i i ≤ ≤ , are the sides of these graphs.Note that

⊕
. It is easy to see that K q 4 and K ˆq 4 are 2  -connected for all q 2 ≥ , and the largest 2  -circuits in these graphs are the subgraphs isomorphic to K K ⊕ .
We are ready to prove the main result of this section.
Proof of Theorem 1.3.We start the proof with a simple observation: by adding a new edge to K q 4 , with q 3 ≥ , we obtain either K ˆq 4 , or a graph which contains an 2  -circuit with eight vertices.To see this consider a subgraph H of K q 4 isomorphic to K 4 3 with sides ⊕ .Therefore we may assume that G is minimally 2  -connected.
Let us suppose that (ii) does not hold.For a contradiction let us also suppose that every 2  -circuit in G has at most six vertices.Let H be a subgraph of G which is either isomorphic to K q 4 , for some q 3 ≥ , or is an 2  -circuit, and for which   V H ( ) is as large as possible.

Let us fix an ear-decomposition
be the 2 circuits induced by the C i 's.By Lemma 3.1 such an ear-decomposition exists and we may

∩
. The minimality of G and Lemma 4.5 imply that Y ≠ ∅.Since H 2 and H 1 share an edge, we have   X 2 ≥ .We shall consider several subcases, depending on the size of Y and X .
First suppose that   Y = 1.By Lemma 5.2 we have Y v = { } for some vertex v of degree three in H 2 .If v is connected to two adjacent vertices u v , of H 1 then G contains an 2 circuit that can be obtained from H 1 by a 1-extension on uv, which contradicts the maximality of H H = 1 .If v has three pairwise nonadjacent neighbours in H 1 then we have which contradicts the maximality of H .So we may assume that   and it is again easy to check that H H So we may assume that Y has two neighbours x y , in H 1 , which means that Y and its two neighbours induce a , then Proposition 5.1(c) shows that either there is an edge uv E H ( ) ⊕ and the neighbours of Y are the common vertices of the two K 4 's.In the former case G contains the 2  -circuit

⊕
, while in the latter case G contains a K 4 3 .Each of these situations contradicts the maximality of [ ] induces a cycle (resp., a path) on three vertices and we have four (resp., five) edges between Y and X by Lemma 5.2.A quick case analysis shows that if X induces an edge e of H 1 then G contains an 2  -circuit which can be obtained from H 1 by three 1-extensions.
If X induces no edge of holds, in which case it is again not hard to check, by using 1-extensions, that G contains an 2  -circuit on nine vertices.In both cases we have a contradiction.Case 2. H K = q 4 for some q 3 ≥ .Let us fix an ear-decomposition −1 is an ear-decomposition of H consisting of q − 1 subgraphs isomorphic to K K

⊕
. It follows from Lemma 3.1 that such an ear-decomposition exists.Since (ii) does not hold by our assumption, H q exists (q t = may hold).
. The minimality of G and Lemma 4.5 imply that Y ≠ ∅.First suppose that   Y = 1.By Lemma 5.2 we have Y v = { } for some vertex v of degree three in H q .If v is connected to two adjacent vertices u w , of H then G contains the 2 circuit that can be obtained from K K ⊕ by a 1-extension on uw.Otherwise v has three pairwise nonadjacent neighbours in H from three different sides of H .In this case it is easy check that G contains an 2  -circuit which can be obtained from K e + 3,3 by a 1-extension.Both alternatives contradict our assumption on the size of a largest 2  -circuit in G.
Next suppose that   X = 2. Let X x y = { , }.We must have xy E H E H ( ) ( ) . Since the 2-sum of two 2  -circuits is an 2  -circuit, we obtain that K K H ( ) circuit of G, a contradiction.So we may assume that   X 3 ≥ and   Y 2 ≥ .If H q intersects at most two sides of H , then we can apply the argument (in the second last paragraph of the proof) of Case 1 to the 2  -circuit of H that contains the two sides to deduce that either we have an 2  -circuit on at least seven vertices or a subgraph isomorphic to K q 4 +1 , a contradiction.Finally, consider the case when H q intersects at least three sides of H .By Lemma 5.2, X is critical in H q and every edge induced by X in H q is an edge of H . Since a critical set on at least three vertices induces a 2-connected graph, a b X { , } ⊂ follows.Therefore which shows that H q is an 2  -circuit on at least seven vertices.This contradiction completes the proof.□ The largest 2  -circuits in the complete bipartite graphs K r 3, , for , obtained from K q 4 by replacing the K 4 's by K d+2 's, show that this bound would be tight.
We can now sharpen Theorem 4.6 and verify the second part of Conjecture 4.2 for d = 2.
Proof of Theorem 1.2.By Theorem 1.3 either G is isomorphic to K q 4 for some q 3 ≥ , or G contains an 2  -circuit H on at least seven vertices.In the former case we have

≤
, with equality for   V = 8.In the latter case we can take an ear-decomposition of G starting with H and apply Theorem 4.3 to get

| MINIMALLY BRIDGELESS MATROIDS AND REDUNDANTLY RIGID GRAPHS
We say that a bridgeless matroid  on ground-set E is minimally bridgeless if e −  has a bridge for all e E ∈ .The bridgeless counterpart of Theorem 1.1 is easier to prove.
Theorem 6.1.Let  be a minimally bridgeless matroid on ground-set E with rank function r and let C be a circuit of .Then . This bound is almost tight for d = 1, but it can probably be improved 2 for all d 2 ≥ .In the rest of this section we determine the best possible upper bound for d = 2 by using our new bound on the size of minimally 2  -connected graphs.The following lemma, which is valid in all dimensions, is a key observation.
Lemma 6.2.Let G V E = ( , ) be a minimally redundantly rigid graph in d and let Proof.We use the fact that a graph G with at least two edges is redundantly rigid in d if and only if G is rigid in d and every edge of G belongs to an d  -circuit of G.This implies that J is a nontrivial d  -component of G. Suppose, for a contradiction, that J e − is d  - connected for some e F ∈ .Since G is redundantly rigid, G e − is rigid.Moreover, every edge of G e − belongs to an d  -circuit of G e − , as J e − is d  -connected and for an edge f E F − ∈ no d  -circuit in G contains both e and f .Therefore G e − is also redundantly rigid in d , contradicting the minimality of G. □ For d = 2 we can deduce a tight upper bound.
Proof of Theorem 1.4.Let J J J , , …, q 1 2 be the 2  -components of G.If q = 1 then we can use Lemma 6.2, Theorem 1.2, and the assumption   V 7 ≥ to obtain (2).Next suppose that q = 2. Since G is rigid, it is 2-connected.Hence J 1 and J 2 share at least two vertices.On the other hand, the fact that the union of two 2  -connected graphs with at least two vertices in common is 2  -connected (see [9,Exercise 3.3.12])implies that G J J = 1 2 ∪ is 2  -connected, a contradiction.So we may assume that q 3 ≥ .Since the rank of a matroid is equal to the sum of the ranks of its connected components, we have 2 Indeed, a recent paper of Király [12] provides almost tight upper bounds for all d 1 ≥ .

JORDÁN | 465
By using Lemma 6.2, Theorem 4.6 and that the 2  -components are pairwise edgedisjoint rigid subgraphs, we obtain Now ( 11), (12), and , as required.□ The minimally redundantly rigid graphs K n 3, −3 for n 7 ≥ show that the bound in Theorem 1.4 is the best possible.Note that for n = 9 another extremal graph can be obtained by replacing each edge of a triangle by a K 4 .

| CONCLUDING REMARKS
The main results of this paper provide tight upper bounds on the number of edges in minimally 2  -connected and minimally redundantly rigid graphs in 2 , respectively.These bounds and their proofs improve on and complete some preliminary observations of the lecture note [9, Section 3.6.1].We also have new upper bounds in the d-dimensional case, for all d 3 ≥ , but these bounds are far from   d V ( + 1) , which is the conjectured extremal number in a graph on   V vertices (up to some constant depending on d).
We remark that the tight upper bounds have been determined on the number of edges in minimally globally rigid and minimally vertex-redundantly rigid graphs in d , as well as for minimally d ( + 1)-connected graphs, for all d 1 ≥ , see [5,11,14].In each of these three families the bound for a graph on   V vertices is minus some constant.It is straightforward to deduce upper bounds on the minimum degree of minimally d  -connected graphs and minimally redundantly rigid graphs in d , respectively, from our new bounds on their size.However, these bounds are not the best possible, even in 1 .Interestingly, better bounds on the minimum degree can be obtained by using eardecompositions, see, for example [8,19].
It follows from Laman's theorem [13] that the matroid G ( ) 2  of a graph G is equal to its count matroid G ( ) k l ,  with parameters k l = 2, = 3. (See [6] for the definition of these matroids, which are also defined on edge sets of graphs.)Our results on connected two-dimensional rigidity matroids can be extended to connected count matroids with arbitrary parameters k k l 1, 2 − 1 0 ≥ ≥ ≥ , by using (strict) ear-decompositions.The key property shared by these matroids is that the subgraphs induced by the edge sets of their circuits are "rigid" and adding edges to the graph preserves the connectivity of the matroid.See [6] for more details.

1 1 4 1. 1 2
and   C = The case i = 2 follows from Claim 3.4, applied to C C , , and the other potential circuits in  D 2  .
minimally connected.Thus we can use Theorem 1.1 to obtain an upper bound on the size of a minimally d  - connected graph G V E = ( , ).It is well known that the minimum degree of an d  -connected graph is at least d + 1.Thus we have case Theorem 1.1 can be applied), or d = 1 and G K = 3 .Moreover,   V d + 2 ≥ and

Theorem 4 . 3 .
Let G V E = ( , ) be an d  -connected graph and suppose that G

3 , 3 ., 4 n≥
The wheel graph W n , is obtained from a cycle of length n − 1 by adding a new vertex v and new edges from v to every other vertex.See Figure4.

FF
I G U R E 3 The double banana graph plus two edges vx vy , .
) be an 2  -circuit with   V 6 ≤ .Then (a) G ∈ , (b) G has no three pairwise nonadjacent vertices, unless G K e for every nonadjacent vertex pair x y V , ∈there is an edge uv Suppose that t 2 ≥ holds.By Lemma 4.4 we must have Y t ≠ ∅.Moreover, Lemma 3.2(ii) implies that the subgraph G′ of G induced by C 2.The statement of the lemma is obvious for t = 1. holds.
degree three in H 2 by Lemma 5.2.Suppose that Y has three or four neighbours in H 1 .If the neighbours of Y induce an edge e of H 1 then it is easy to see that H H  -circuit which can be obtained from H 1 by two 1-extensions, so that the first one is made on edge e, and the second one is made on a new edge.It contradicts the maximality of H .If the neighbours of Y induce no edges of H 1 then H r 4 ≥ , are the subgraphs isomorphic to K 3,4 .Thus the bound in Theorem 1.3(i) is tight.
) has a circuit of size at least four.The graphic matroids of the graph obtained from a path on n vertices by replacing each edge by two parallel edges, and that of the graph K n ∪be a base of , where r r E = ( ).Let f E ∈ be the unique element