characterizing 3-connected planar graphs and graphic matroids

Characterizing3-Connected PlanarGraphs and GraphicMatroids

Manoel Lemos,1 Talmage James Reid,2 and Haidong Wu2

1DEPARTAMENTO DE MATEMATICAUNIVERSIDADE FEDERAL DE PERNAMBUCO

RECIFE, PERNAMBUCO, BRAZILE-mail: [email protected]

2DEPARTMENT OF MATHEMATICS UNIVERSITYOF MISSISSIPPI, MISSISSIPPI

E-mail: [email protected], [email protected]

Received May 6, 2008; Revised June 18, 2009

Published online 25 August 2009 in Wiley InterScience (www.interscience.wiley.com).DOI 10.1002/jgt.20446

Abstract: A well-known result of Tutte states that a 3-connected graphG is planar if and only if every edge of G is contained in exactly two inducednon-separating circuits. Bixby and Cunningham generalized Tutte’s resultto binary matroids. We generalize both of these results and give new char-acterizations of both 3-connected planar graphs and 3-connected graphicmatroids. Our main result determines when a natural necessary condition

Contract grant sponsor: CNPq; Contract grant numbers: 301178/05-4;502048/07-7; 48567/07-7; Contract grant sponsor: FAPESP/CNPq; Contract grantnumber: 2003/09925-5.

Journal of Graph Theory� 2009 Wiley Periodicals, Inc.

165

166 JOURNAL OF GRAPH THEORY

for a binary matroid to be graphic is also sufficient. � 2009 Wiley Periodicals, Inc. J

Graph Theory 64: 165–174, 2010

Keywords: 3-connected planar graphs; graphic matroids; non-separating cocircuits; binary

matroids

1. INTRODUCTION

Non-separating cocircuits in matroids have been extensively studied and have importantapplications to graph planarity questions (see [1–7, 10, 12]). Well-known results ofHalin and Mader on vertices in graphs are generalized to binary matroids using thesecocircuits [12]. A cocircuit D in a connected matroid M is non-separating if and onlyif M\D is connected. If M is a 3-connected matroid that is the cycle matroid of agraph G, then a set Z is a non-separating cocircuit of M if and only if Z is the set ofedges incident with a vertex of G. Thus every edge of G is in the two non-separatingcocircuits of M corresponding to its end-vertices. Thus the following natural necessarycondition for a 3-connected binary matroid to be graphic holds.

(P). Let M be a 3-connected binary matroid with rank exceeding one. If M isgraphic, then any k-element (edge) set A of M meets at most 2k non-separating cocircuitsof M.

The next result of Bixby and Cunningham [1] showed that the converse of (P) holdswhen k=1.

Theorem 1.1. If M is a 3-connected binary matroid with r(M)≥2, then M is graphicif and only if every element of M belongs to at most two non-separating cocircuits.

This result motivates the following question.

Question. Is the converse of (P) true when k exceeds one?

The main result of the paper is given next. It answers the question affirmatively fork≤3. This result is almost best possible as we show that the answer to the questionis negative for k=4 after the statement of the theorem, and, for k≥6 by presentingexamples of arbitrarily high rank in the last section of the paper.

Theorem 1.2. If M is a 3-connected binary matroid with r(M)≥2, then the followingstatements are equivalent:

(i) M is graphic.(ii) Each element of M meets at most two non-separating cocircuits.

(iii) Each pair of elements of M meets at most four non-separating cocircuits.(iv) Each set of three elements of M meets at most six non-separating cocircuits.

Note that the Fano matroid F7 has exactly seven non-separating cocircuits. Therefore,part (i) of the theorem is not equivalent to the assertion that each set of four elements

Journal of Graph Theory DOI 10.1002/jgt

3-CONNECTED PLANAR GRAPHS AND GRAPHIC MATROIDS 167

of M meets at most eight non-separating cocircuits. The above result does not useTheorem 1.1 in its proof and clearly generalizes the theorem also.

If A is a subset of the ground set of a 3-connected binary matroid M, then we say thatA is an obstruction for graphicness provided that A meets at least 2|A|+1 non-separatingcocircuits of M. Theorem 1.1 says that a non-graphic 3-connected binary matroid has aset of obstruction for graphicness of size one. Our result studies the set of obstructionfor graphicness of larger size. This kind of obstruction for graphicness will providemore insight in studying the structure of non-graphic binary matroids. Therefore, ourmain question is not only a natural question to study, but also an important one.

For a 2-connected graph G, a circuit C is non-separating if and only if the contractionG /C is 2-connected. A circuit C is an induced non-separating circuit of G if and onlyif C is a non-separating cocircuit of M∗(G). Therefore, the following result of Tutte[10], a fundamental characterization of planar graphs, is an immediate consequence ofTheorem 1.1 and therefore our result.

Theorem 1.3. Let G be a simple 3-connected graph. Then G is planar if and only ifevery edge meets at most (in fact, exactly) two induced non-separating circuits.

We obtain the following new graph planarity condition as a consequence ofTheorem 1.2 by letting M be a cographic matroid. This result generalizes Tutte’stheorem.

Theorem 1.4. If M is a 3-connected simple graph, then the following statements areequivalent:

(i) G is planar.(ii) Each edge of G meets at most two induced non-separating circuits.

(iii) Each pair of edges of G meets at most four induced non-separating circuits.(iv) Each set of three edges of G meets at most six induced non-separating

circuits.

We are able to obtain a sharper result than Theorem 1.2 when restricting our attentionto the class of regular matroids.

Theorem 1.5. If M is a 3-connected regular matroid with r(M)≥2, then the followingstatements are equivalent:

(i) M is graphic.(ii) Each element of M meets at most two non-separating cocircuits.

(iii) Each pair of elements of M meets at most four non-separating cocircuits.(iv) Each set of three elements of M meets at most six non-separating cocircuits.(v) Each set of four elements of M meets at most eight non-separating cocircuits.

The proofs of Theorems 1.2 and 1.5 are given in Sections 2 and 3, respectively.The terminology follows [8] with the following exceptions. Let M be a matroid. Weuse co(M\e) and si(M /e) to denote the cosimplification and simplification of M\e andM /e, respectively. We use R∗(M) to denote the set of non-separating cocircuits of M.The set of edges meeting a vertex v of a graph G is denoted by st(v).



2. A PROOF OF THE MAIN RESULT

We present a proof of the main result after first giving some preliminary lemmas.Whittle [11, Lemma 3.4] gave the following easy consequence of Seymour’s SplitterTheorem [9].

Lemma 2.1. Suppose that N is a 3-connected minor of a 3-connected matroid M. Ifr∗(M)>r∗(N), then there is an element e of E(M) such that co(M\e) is 3-connectedwith an N-minor.

The next lemma is due to Lemos [4].

Lemma 2.2. Suppose that e is an element of a 3-connected binary matroid M suchthat co(M\e) is 3-connected. Then it is possible to choose the ground set of co(M\e) sothat, for each C∗ ∈R∗(co(M\e)), C∗�X ∈R∗(M), where X =∅, X ={e}, or X =T∗−e,for some triad T∗ meeting both e and C∗.

Lemma 2.3. Let A be a k-element subset (k≥2) of a matroid M. If A meets at least2k+1 non-separating cocircuits, then A has a (k−1)-element subset that meets at least2k−1 non-separating cocircuits.

Proof. Let S be a set of 2k+1 non-separating cocircuits that each meet A. Supposethat every (k−1)-element subset A−a of A meets at most 2k−2 members of S. Theneach element a of A is in at least three members of S that meet A only in a. Hence2k+1=|S|≥3|A|=3k so that k≤1, a contradiction. �

Lemma 2.4. Suppose that e is an element of a 3-connected binary matroid M suchthat co(M\e) is 3-connected. For an integer k≥3, if E(co(M\e)) has a k-element subsetmeeting at least 2k+1 non-separating cocircuits of co(M\e), then E(M) has a k-elementsubset meeting at least 2k+1 non-separating cocircuits of M.

Proof. Suppose the result does not hold. By Lemma 2.2, it is possible to choosethe ground set of co(M\e) so that, for each C∗ ∈R∗(co(M\e)), C∗�X ∈R∗(M), whereeither X =∅, X ={e}, or X =T∗−e, for some triad T∗ meeting both e and C∗. Assumethat A is a k-element subset of E(co(M\e)) meeting the non-separating cocircuitsC∗

1,C∗2, . . . ,C∗

2k+1 of co(M\e)). For each i∈{1,2, . . . ,2k+1}, let Xi be as above suchthat C∗

i �Xi is a non-separating cocircuit of M.For each a∈A, let T∗

a ={e,a,a′} be the unique triad of M that contains both e and a,when it exists. If A∩(C∗

i �Xi)=∅, then A∩C∗i ⊆Xi and so |A∩C∗

i |=1, say A∩C∗i =

{ai}. In this case, T∗ai

exists and Xi =T∗ai

−e. For each a∈A, set Ia ={i∈{1,2, . . . ,2k+1} :ai exists and ai =a}.

Each element of the next family of non-separating cocircuits of M meets the setB′ ={a′ :a∈B}, where B={a∈A : |Ia|≥2}:

�= ⋃

a∈B({C∗

i �Xi : i∈ Ia}∪{T∗a }).

Note that |B|=|B′|. Observe that

|�|=|B′|+ ∑

a∈B|Ia|≥3|B′|. (1)



Define also

�={C∗i �Xi : i∈{1, . . . ,2k+1} and ai does not exist}∪{T∗

a :a∈A and |Ia|=1}.Then each element of � is a non-separating cocircuit of M that meets A. Moreover,

|�|= (2k+1)− ∑

a∈B|Ia|. (2)

It follows from (1) and (2) that

|�|= (2k+1)+|B′|−|�|. (3)

If B′ =∅, then �=∅ and so � is a family with 2k+1 non-separating cocircuits of Msuch that each one of them meets A; a contradiction. Therefore B′ �=∅.

Choose A′ ⊆A and �′ ⊆� such that

|�′|≥2|A′|, (4)

every element of �′ meets A′, |A′|≤k−|B′|, and |A′|+|�′| is maximal. (Note that A′and �′ exist because (A′,�′)= (∅,∅) satisfies the first three conditions.) Observe thatevery element of �−�′ meets A−A′.

Now, we prove that

|A′|<k−|B′|. (5)

If (5) does not hold, then, by the choice of A′, |A′|+|B′|=k. By (1) and (4),

|�′ ∪�|≥2|A′|+3|B′|=|B′|+2(|A′|+|B′|)=|B′|+2k≥2k+1.

Note that each element of �′ ∪� meets A′ ∪B′, a contradiction. Hence (5) follows. By(3), we may assume that �′ �=�; otherwise, �′ ∪� is a family with at least 2k+1non-separating cocircuits of M with each one of them meeting A′ ∪B′.

Suppose that a∈A−A′. If D∗1,D∗

2, . . . ,D∗l are the elements of �−�′ that contain a,

then, by the choice of (A′,�′),

|�′ ∪{D∗1, . . . ,D∗

l }| < 2|A′ ∪a||�′|+ l ≤ 2|A′|+1

By (4), l≤1 and, when l=1, we have equality in (4). So the equality in (4) followsbecause � �=�′, and thus there is an element of A−A′ meeting a cocircuit belongingto �−�′. Moreover, by the Pigeonhole Principle,

|�|−|�′|=|�−�′|≤|A−A′|=|A|−|A′|.As equality holds in (4), it follows that

|�|≤|A|+|A′|=k+|A′|. (6)

By (3) and (6), we obtain

|�|≥ (2k+1)+|B′|−(k+|A′|)=k+1+|B′|−|A′|.By the equality that occurs in (4), |�∪�′|≥k+1+|B′|+|A′| and so

(2k+1)−|�∪�′|≤k−|A′∪B′|. (7)



As every element of �∪�′ meets A′ ∪B′, it follows, by (7), that we can choose a family� with (2k+1)−|�∪�′| non-separating cocircuits of M disjoint from �∪�′ and a(k−|A′∪B′|)-element subset A′′ of E(M) disjoint from A′ ∪B′ that meets each elementof �. Note that �∪�′ ∪� is a family of non-separating cocircuits of M having 2k+1elements such that each one of its elements meets A′ ∪A′′ ∪B′ which is a k-subset ofE(M), a contradiction. The result follows. �

Now we are ready to prove our main theorem.

Proof of Theorem 1.2. Note that (i) implies (ii), and (ii) implies (iii). By Lemma 2.3,we deduce that (iii) implies (iv). Thus we need to only show that (iv) implies (i).That is, Theorem 1.1 is true under weaker conditions (iii) or (iv).

Now we show that (iv) implies (i). Let M be a minimal counterexample. Then M is3-connected, non-graphic, binary, and every 3-element subset of E(M) meets at mostsix non-separating cocircuits of M. We first establish that M is regular. If M is notregular, then M contains F7 or F∗

7 as a minor. It follows from counting the numberof non-separating cocircuits meeting any set of three independent points of F7, andmeeting any triad of F∗

7 , that M is isomorphic to neither F7 nor F∗7 . Then F7 is a

splitter of the class of 3-connected binary matroids without F∗7-minors [9, (7.6)] so

that M has an F∗7- minor. By Lemma 2.1, M has an element e such that co(M\e) is

3-connected and has an F∗7-minor. By the choice of M, there is a 3-element subset A of

E(co(M\e) that meets at least seven non-separating cocircuits of co(M\e). We arriveat a contradiction by Lemma 2.4. Therefore M is regular.

The matroid M is not isomorphic to M∗(K5) because every 3-element subset ofM∗(K5) meets at least seven triangles of K5 (The triangles of K5 are the non-separatingcocircuits of M∗(K5)). It follows from [9, (7.5)] that M∗(K5) is a splitter for the class ofregular matroids without a M∗(K3,3)-minor so that M has an M∗(K3,3)-minor. Now, weprove that M and M∗(K3,3) have the same corank. Suppose not. By Lemma 2.1, M hasan element e such that co(M\e) is a 3-connected matroid having an M∗(K3,3)-minor.By the choice of M, there is a 3-element subset A of E(co(M\e)) that meets at leastseven non-separating cocircuits of co(M\e). We arrive at a contradiction by Lemma2.4. Thus r∗(M)=r∗(M∗(K3,3)). Hence M is cographic with M =M∗(G) for some graphG. There is a set X of edges of G such that G\X =K3,3 because r∗(M)=r∗(M∗(K3,3)).Every triangle T of G that contains just one edge in X is a non-separating cocircuitof M. Note that each edge of X belongs to exactly three of these triangles. Therefore|X|≤1, and M is isomorphic to M∗(K3,3) or M∗(K′

3,3), where K′3,3 is the graph obtained

from K3,3 by adding an edge in a partite set of size three. In both cases, M has a3-element set meeting at least seven non-separating cocircuits of M, a contradiction.This completes the proof of the theorem. �

3. THE REGULAR MATROID CASE

In this section, we extend our main theorem for regular matroids. Recall that A isan obstruction for graphicness provided that A meets at least 2|A|+1 non-separating



cocircuits of M. If M is non-graphic, then we define

b(M)=max{|A| :A is an obstruction for graphicness}.Let M be a matroid and N =M /e for some element of M. Then M is called a lift

of N. Let F be a class of binary matroids closed under minors and isomorphisms. Fora 3-connected matroid N belonging to F , we denote by FN the class of 3-connectedcorank-preserving lifts of N belonging to F . We define

�F (N)=min{b(M) :M ∈FN}.For example:

�B(F∗7)=3 and �R(M∗(K3,3))=4,

where B and R are, respectively, the class of binary and regular matroids. We have thefollowing result:

Lemma 3.1. Let N and M be 3-connected matroids belonging to F such that r(N)≥3.If N is a minor of M, then b(M)≥�F (N).

Proof. By induction on r∗(M), if r∗(M)=r∗(N), then the result follows by defini-tion. Assume that r∗(M)>r∗(N). By Lemma 2.1, M has an element e such that co(M\e)is a 3-connected matroid having N as minor. By induction, b(co(M\e))≥�F (N). Theresult follows, by Lemma 2.4, because b(M)≥b(co(M\e)). �

As a consequence of this result we have that:

Corollary 3.2. If M is a 3-connected regular matroid, then M is graphic if and onlyif each 4-element subset of E(M) meets at most eight non-separating cocircuits.

Proof. It is enough to show that, when M is non-graphic, E(M) has a 4-elementsubset which is an obstruction for graphicness—that is, we need to establish that

b(M)≥4. (8)

Observe that (8) holds when M ∼=M∗(K5). We may assume that M �∼=M∗(K5). By[9, (7.5)], M has a minor isomorphic to M∗(K3,3). Thus (8) follows because, byLemma 3.1, b(M)≥�R(M∗(K3,3))=4. �

Note that Theorem 1.5 follows immediately from Theorem 1.2 and the abovecorollary.

4. A COUNTEREXAMPLE

In this section, we provide a negative answer to our main question for all k≥6. Letv1, v2, v3 be the vertices of degree n, for n≥4, of K3,n. Choose a vertex v of degreethree of K3,n. For each i∈{1,2,3}, let ei be the edge of K3,n incident to both vi and v.



Let e be an element not belonging to E(K3,n). Assume that M is the binarymatroid over E(K3,n)∪e such that M\e=M(K3,n) and C={e1,e2,e3,e} is a circuitof M.

If C∗ is a cocircuit of M that contains e, then C∗−e is a cocircuit of M\e=M(K3,n).So C∗−e is a bond of K3,n. Therefore C∗ is a non-separating cocircuit of M if and onlyif C∗−e is a non-separating cocircuit of M(K3,n); this is true if and only if C∗−e is thestar of some vertex of K3,n. By orthogonality this vertex must be v1, v2, v3 or v. Thusthere are exactly 4 non-separating cocircuits of M that contain e, namely: st(v1)∪e,st(v2)∪e,st(v3)∪e and st(v)∪e.

Let w /∈{v,v1,v2,v3} be a vertex of K3,n. Then st(w) is a triad of M\e; hence, eitherst(w) or st(w)∪e is a cocircuit of M. By orthogonality, the latter cannot happen as{e,e1,e2,e3} is a circuit. We conclude that st(w) is a (non-separating) triad of both Mand M\e.

Let C∗ be a non-separating cocircuit of M avoiding e. As C∗ does not span e inM∗, it follows that C∗ is a cocircuit of M\e=M(K3,n). There is a partition {X,Y}of V(K3,n) such that C∗ is the set of edges of K3,n incident with some vertex inX and some vertex in Y . Assume that v∈Y . By orthogonality, |C∗∩C| is even.If C∗∩C=∅, then {v1,v2,v3}⊆Y and so E(K3,n[Y]) spans e in M. Thus |X|=1.That is, C∗ =st(w), for some w∈V(K3,n)−{v,v1,v2,v3}. Now, suppose that |C∗∩C|=2, say {ei,ej}=C∗∩C and {e,ek}=C−C∗, where {i, j,k}={1,2,3}. Therefore{vi,vj}⊆X and vk ∈Y . Note that X−{vi,vj} �=∅, say x∈X−{vi,vj} (otherwise C∗ =st(vi)∪st(vj), which is not a cocircuit of K3,n, a contradiction). As {xvi,viv,vvj,vjx}is a circuit of M(K3,n), it follows that D=C�{xvi,viv,vvj,vjx}={xvi,xvj,vvk,e} is acircuit of M. Now we prove that Y ={v,vk}. If w∈Y −{v,vk}, then vkw is a coloopof M(K3,n)\C∗ =M\C∗\e. Hence either vkw or {vkw}∪e is a cocircuit of M\C∗.Note that E(K3,n[X])∪vkv spans e in M. By orthogonality, {vkw}∪e is not a cocircuitof M\C∗. We conclude that vkw is a coloop of M\C∗; a contradiction. Thus Y ={v,vk} and so C∗ =st(v)�st(vk). On the other hand, it is straightforward to verify thatC∗ =st(v)�st(vk) is non-separating in M. In summary, the non-separating cocircuits ofM avoiding e are: C∗ =st(w), for some w∈V(K3,n)−{v,v1,v2,v3},st(v)�st(v1),st(v)�st(v2) and st(v)�st(v3).

Next we show that each six-element subset of E(M) meets at most 12 non-separatingcocircuits of M. Indeed, let X be a 6-element subset of E(M). First, we show that Xmeets at most nine of the non-separating cocircuits in the family

{st(w) :w∈V(K3,n)−{v,v1,v2,v3}}∪{st(w)∪{e} :w∈{v,v1,v2,v3}}. (9)

We have two cases to consider. If e �∈X, then X meets st(w)∪Z, where Z ⊆{e} ifand only if X meets st(w) and if and only if w∈V(X). But |V(X)|≤9 and X meetsat most nine cocircuits belonging to the family given in (9). Assume now that e∈X.Thus X meets every cocircuit in {st(w)∪{e} :w∈{v,v1,v2,v3}}. Now, X meets st(w), forw∈V(K3,n)−{v,v1,v2,v3} if and only if w∈V(X). As |V(X)∩[V(K3,n)−{v,v1,v2,v3}]|≤|X−e|≤5, it follows that X meets at most nine cocircuits belonging to the family givenin (9). As M has only three other non-separating cocircuits not belonging the familygiven in (9), it follows that X meets at most 12 non-separating cocircuits of M.



For any set A with |A|≥7, by Lemma 2.3, A meets at least most 2k non-separatingcocircuits of M. Since M is not graphic, this matroid is a counterexample to our mainquestion for all k≥6.

5. CONCLUDING REMARKS

Non-separating cocircuits are natural generalizations of vertices for binary matroids.Tutte and Bixby and Cunningham’s results show that non-separating cocircuits are veryuseful in characterizing 3-connected planar graphs and graphic matroids. Therefore, itis important to further study non-separating cocircuits for binary matroids. Our mainresult answers a natural question for k at most three by showing that a natural necessarycondition turns out to be also sufficient for a 3-connected binary matroid to be graphic.Our result also generalizes both Tutte’s and Bixby and Cunningham’s results, and isalso almost best possible in the sense that it is not true for k=4 and for k≥6. Moreover,our result studies the set of obstruction for graphicness of larger size. We hope that thiskind of obstruction for graphicness will provide more insight in studying the structureof non-graphic binary matroids.

REFERENCES

[1] R. E. Bixby and W. H. Cunningham, Matroids, graphs, and 3-connectivity,Graph Theory and Related Topics (Proc. Conf., Univ. Waterloo, Waterloo,Ont., 1977), Academic Press, New York, 1979, pp. 91–103.

[2] B. M. Junior, M. Lemos, and T. R. B. Melo, Non-separating circuits andcocircuits in matroids, Combinatorics, Complexity, and Chance, OxfordLecture Series in Mathematics and its Applications 34, Oxford UniversityPress, Oxford, 1993, pp. 162–171.

[3] A. K. Kelmans, Graph planarity and related topics, Graph StructureTheory (Seattle, WA, 1991), Contemporary Mathematics 147, AmericanMathematical Society, Providence, RI, 1993, pp. 635–667.

[4] M. Lemos, Non-separating cocircuits in binary matroids, Linear AlgebraAppl 382 (2004), 171–178.

[5] M. Lemos, A characterization of graphic matroids using non-separatingcocircuits, Adv Appl Math 42 (2009), 75–81.

[6] M. Lemos, T. J. Reid, and H. Wu, Obstructions to a binary matroid beinggraphic, submitted.

[7] J. McNulty and H. Wu, Connected hyperplanes in binary matroids, J CombinTheory Ser B 79(1) (2000 ), 87–97.

[8] J. G. Oxley, Matroid Theory, Oxford Science Publications, The ClarendonPress, Oxford University Press, New York, 1992.

[9] P. D. Seymour, Decomposition of regular matroids, J Combin Theory Ser B28(3) (1980), 305–359.



[10] W. T. Tutte, How to draw a graph, Proc London Math Soc 13(3) (1963),743–767.

[11] G. Whittle, Stabilizers of classes of representable matroids, J Combin TheorySer B 77(1) (1999), 39–72.

[12] H. Wu, On vertex-triads in 3-connected binary matroids, Combin ProbabComput 7(4) (1998), 485–497.


characterizing 3-connected planar graphs and graphic matroids

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