DOI: 10.1051/ita:2005005
ON THE NUMBER OF DISSIMILAR PFAFFIAN ORIENTATIONS OF GRAPHS
∗Marcelo H. de Carvalho
1, Cl´ audio L. Lucchesi
2and U.S.R. Murty
3Abstract. A subgraphH of a graphGisconformalifG−V(H) has a perfect matching. An orientationD of Gis Pfaffian if, for every conformal even circuit C, the number of edges of C whose directions in D agree with any prescribed sense of orientation of C is odd. A graph is Pfaffian if it has a Pfaffian orientation.
Not every graph is Pfaffian. However, ifGhas a Pfaffian orienta- tionD, then the determinant of the adjacency matrix ofDis the square of the number of perfect matchings of G. (See the book by Lov´asz and Plummer [Matching Theory. Annals of Discrete Mathematics, vol. 9. Elsevier Science (1986), Chap. 8.] Amatch- ing covered graphis a nontrivial connected graph in which every edge is in some perfect matching. The study of Pfaffian orien- tations of graphs can be naturally reduced to matching covered graphs. The properties of matching covered graphs are thus help- ful in understanding Pfaffian orientations of graphs. For example, say that two orientations of a graph aresimilar if one can be ob- tained from the other by reversing the orientations of all the edges in a cut of the graph. Using one of the theorems we proved in [M.H. de Carvalho, C.L. Lucchesi and U.S.R. Murty, Optimal ear decompositions of matching covered graphs. J. Combinat. The- ory B85(2002) 59–93] concerning optimal ear decompositions, we show that if a matching covered graph is Pfaffian then the number of dissimilar Pfaffian orientations ofGis 2b(G), whereb(G) is the number of “bricks” ofG. In particular, any two Pfaffian orienta- tions of a bipartite graph are similar. We deduce that the problem of determining whether or not a graph is Pfaffian is as difficult as
∗ Supported by CNPq, Brasil, by PRONEX/CNPq (664107/1997-4), by FUNDECT-MS (0284/01) and by a Fellowship for the first author from the University of Waterloo, Canada.
1ufms, Campo Grande, Brasil and University of Waterloo, Waterloo, Canada.
2unicamp, Campinas, Brasil;lucchesi@ic.unicamp.br
3University of Waterloo, Waterloo, Canada.
c EDP Sciences 2005
the problem of determining whether or not a given orientation is Pfaffian, a result first proved by Vazirani and Yanakakis [Pfaffian orientation of graphs, 0,1 permanents, and even cycles in digraphs.
Discrete Appl. Math. 25 (1989) 179–180]. We establish a simple property of minimal graphs without a Pfaffian orientation and use it to give an alternative proof of the characterization of Pfaffian bipartite graphs due to Little [A characterization of convertible (0,1)-matrices. J. Combinat. Theory B18(1975) 187–208].
Mathematics Subject Classification. 05C70
1. Introduction
Let A = (aij) be an n×n skew-symmetric matrix. When n is odd, then it is easy to see that det(A) = 0. On the other hand, whenn is even, there is a polynomialP :=P(A) in theaij such that detA=P2. This polynomial is called thePfaffian ofAand is defined as follows:
P :=
sgn(M)ai1j1ai2j2...aikjk, (1) where the sum is taken over the set of all partitionsM = (i1j1, i2j2, ..., ikjk) of {1,2, ..., n}intokunordered pairs, and sgn(M) is the sign of the permutation:
π(M) :=
1 2 3 4 . . . 2k−1 k i1 j1 i2 j2 . . . ik jk
.
It can be seen that the definition of the Pfaffian ofAgiven above is independent of the order in which the constituent pairs in a partitionM are listed, as also of the order in which the elements in a pair are listed. SinceA is skew-symmetric, for each pair (i, j) of indices, eitheraij or ajiis nonnegative.
Now suppose thatGis a graph,Dis an orientation ofG, andAis the adjacency matrix of D. Then each nonzero term in the expansion of the Pfaffian of A corresponds to a perfect matchingM ofG. Thus, ifDis such that all sgn(M) are the same, then|P|is the number of perfect matchings ofG.
An orientation D of a graph G is a Pfaffian orientation of G if all perfect matchings of G have the same sign. (We shall see below that this is equivalent to the definition given in the abstract.) An undirected graphG is Pfaffian if it admits a Pfaffian orientation.
1.1. Parities of circuits
The parity of a circuit C of even length in a directed graph is the parity of the number of its edges that are directed in agreement with a specified sense of orientation of C. As C has an even number of edges, the parity is the same in both senses and thus is well defined. For any two sets X and Y, we denote by X⊕Y the symmetric difference ofX andY.
Theorem 1.1(see [12], Lem. 8.3.1). LetDbe an arbitrary orientation of an undi- rected graphG. LetM1andM2be any two perfect matchings ofGand letkdenote the number of even parity circuits of G[M1⊕M2]. Then, M1 and M2 have the same sign if and only ifkis even.
Recall that an even circuit C of G is conformal if G−V(C) has a perfect matching. The following theorem may be derived from the above theorem.
Theorem 1.2(see [12], Th. 8.3.2). LetGbe a graph, M a perfect matching ofG andD an orientation ofG. Then the following properties are equivalent:
• D is a Pfaffian orientation of G.
• Every M-alternating circuit ofGhas odd parity.
• Every conformal circuit ofG has odd parity.
1.2. Changing orientations along cuts
For any graphGand any set X of vertices of G, we denote by ∂(X) the cut ofGhavingX as one of itsshores, that is, the set of edges ofGthat have precisely one end inX. The following simple result is fundamental for most of our results in this paper.
Proposition 1.3. In any graphG, a setRof edges ofGis a cut ofGif and only if|R∩E(Q)|is even, for each circuitQ ofG.
IfD is an orientation of a graphGandS⊂E(G), we denote by DrevGS the orientation ofGobtained from D by reversing the orientation of the edges of S.
WhenGis understood, we only write DrevS instead of DrevGS. We say that two (not necessarily Pfaffian) orientations of a graph are similar if one may be obtained from the other by reversing the edges of some cut of the graph. Observe that changing the direction of an edge corresponds to changing the parity of each circuit of even length containing it. Since every circuit and every cut have an even number of edges in common, it follows that changing the orientations along a cut (that is, reversing the directions of all the arcs in a cut) preserves the parities of the circuits of even length. The following property is a direct consequence of this observation.
Proposition 1.4. Let D be a Pfaffian orientation of a graph G. Then, any orientation ofGthat is similar toD is also Pfaffian.
One might wonder whether the converse of the above proposition is also true;
that is, given two Pfaffian orientations of a graphG, is it possible to start from one of them and reach the other by changing orientations of the edges of some cut?
In Section 3, we answer this question affirmatively for bipartite graphs. More precisely, we shall show that any two Pfaffian orientations of a bipartite graph differ by exactly a cut (that is, they are similar). In fact, we show that for any Pfaffian matching covered graphG, the number of dissimilar Pfaffian orientations ofGis equal to 2b(G), where b(G) denotes the number of bricks ofG.
C
Q Q
(a)
u e v
(b) (c)
Figure 1. The Petersen graph. (a) two disjoint pentagons Q andQ, and 5-cutC; (b) conformal octagonR(e); (c) a conformal octagon containing four edges inC.
We remark that it follows, by Proposition 1.4, that an orientation of any span- ning tree can be extended to a Pfaffian orientation of a Pfaffian graph. Using this observation, plus the fact that every circuit ofK3,3is conformal, it is easy to prove thatK3,3is non-Pfaffian. To illustrate the role of changing orientations along cuts in this theory, we shall now use Proposition 1.4 to show that the Petersen graph is non-Pfaffian.
We say that two orientationsD andD of anuv-pathP have the same parity if the parity of the number of edges directed forward coincide inD andDwhen traversingP from u tov (or from v to u). The following property is easy to be verified.
Proposition 1.5. Let P:= (v0, v1, . . . , vk)be av0vk-path, D1 andD2two orien- tations ofP that have the same parity. Then, for some subsetX of{v1, . . . , vk−1}, D2=D1rev∂(X).
Proposition 1.6. The Petersen graphP is non-Pfaffian.
Proof. LetQbe any pentagon ofP. Then,V(P)−V(Q) spans another pentagon ofP, sayQ (see Fig. 1a).
Assume, to the contrary, thatP has a Pfaffian orientation. We shall now make use of Proposition 1.4 in order to show thatP has a Pfaffian orientation such that each ofQandQ is a directed circuit.
Indeed, given an orientation D of any graph G and a circuit Q of G of odd length, it is easy to see that there exists a setX⊂V(Q) such that reversal of the edges of cut C :=∂(X) yields an orientation D of G that rendersQ a directed circuit. To see this, consider orientation D of G and note that as Q has odd length, one sense of orientation of Q contains an odd number of forward edges and an even number of reverse edges. Letebe a forward edge of Q,uand v its ends. AsT :=Q−e is a path having an even number of forward edges in D, it follows by Proposition 1.5 that there exists a set X ⊂ V(Q)−u−v such that reversal of the edges of cutC:=∂(X) inDproduces an orientationD ofGthat
renders T a directed path, in the same sense of orientation of Q. We conclude that Qis a directed circuit inD. We remark that the shoreX ofC is a subset ofV(Q)−u−v. Therefore, we may in fact apply this reasoning for each member of a collection of vertex-disjoint odd circuits ofG. In particular, in the case of the Petersen graph, the pentagonsQandQare disjoint. We may thus assume thatP has a Pfaffian orientationD such thatQandQ are both directed circuits.
Note that C :=∂(V(Q)) is a 5-cut of P (see Fig. 1a). We now show that C is a directed cut in D. For this, observe first that for any edge e ={uv} of P, P−u−v contains an octagon, which is thus conformal. In particular, for each edgee={uv} ofQ,P−u−vcontains an octagon,R(e), that contains precisely two edges inC, plus two edges inQ, and four edges inQ(see Fig. 1b). Moreover, the restriction of R(e) to each of Q and Q is a directed path, or the reverse of a directed path. Since R(e) must have odd parity, it follows that both edges ofR(e) inCare directed away fromV(Q) or both directed away fromV(Q). This conclusion holds for each edge e of Q. By applying this reasoning for any four edgeseofQ, we conclude thatCis a directed cut inD.
Finally, consider a conformal octagonS that contains four edges inC and two edges in each of Q and Q (see Fig. 1c). The edges of S in Q are traversed in opposite directions, relative to any sense of orientation of Q. Therefore, Q contributes to the parity ofS with an odd number of forward edges. Likewise, so doesQ. Finally, of the four edges ofS in C, as C is directed, two are traversed in a forward manner, two in reverse manner. We deduce thatS has precisely four forward edges, whence it has even parity. AsSis conformal, this is a contradiction.
As asserted,P is non-Pfaffian.
The idea of using Pfaffians in matching theory is due to Tutte. In his book
“Graph Theory As I Have Known It” [15], he describes how he came to the idea of using Pfaffians to find a formula for the number of perfect matchings of a graph.
Although he did not succeed in finding such a formula, Tutte was able to use Pfaffian identities to prove his famous theorem characterizing graphs that have a perfect matching. This was the genesis of the theory of Pfaffian orientations. Sur- prisingly, the Pfaffian orientation problem is related to a number of fundamental, and seemingly unrelated, problems in graph theory. For example, the problem of deciding whether a given directed graph has an even directed circuit is equivalent to deciding if a related bipartite graph has a Pfaffian orientation [16].
Kasteleyn [8] showed that every planar graph has a Pfaffian orientation and pre- sented a polynomial-time algorithm to find such an orientation. The graphK3,3 is the smallest non-Pfaffian graph. Little [9] showed that a bipartite graph is non-Pfaffian if and only if it contains a conformal subgraph that is an even subdi- vision ofK3,3. Thus, the problem of deciding whether a given bipartite graph is Pfaffian is inco− N P. This gave rise to the natural question whether this class of graphs is in N P. Vazirani and Yanakakis [16] showed that it is no easier to decide whether a given orientation of a graph is Pfaffian than deciding that the graph is Pfaffian and finding a Pfaffian orientation if it is. (As we shall see, for bipartite graphs this result follows from the fact that, up to similarity, there is
only one Pfaffian orientation of a Pfaffian bipartite matching covered graph.) We give an alternate proof of this result and show that for general graphs, each of the two problems is polynomially reducible to the other. Finally, Robertson, Seymour and Thomas [14], and independently McCuaig [13], discovered a polynomial-time algorithm for deciding whether or not a bipartite graph has a Pfaffian orientation.
The complexity status of the Pfaffian orientation problem for general graphs is still very much open.
2. Matching covered graphs
A graph is matching covered if it has at least two vertices, is connected and each of its edges lies in a perfect matching. For instance, K3,3, K4, C6 and the Petersen graph are examples of matching covered graphs. We have seen that in the study of Pfaffian orientations we may restrict our attention to matching covered graphs. There is an extensive theory of matching covered graphs, see [11, 12].
We shall assume that the reader is familiar with the basic concepts and results concerning matchings and matching covered graphs. However, we shall review a few basic definitions and results that are relevant to this work. For notation and terminology not defined here, see [3, 4, 11].
2.1. Tight cuts
LetGbe a matching covered graph. A cutC istrivialif either of its shores is a singleton. A cutC is odd(even) if both its shores have odd (even) cardinality.
As |V| is even, then every cut C is either odd or even, and if C is odd, then
|C∩M|is odd for every perfect matchingM ofG. A cutC ofGis atight cutif
|C∩M|= 1 for every perfect matchingM ofG. Trivial cuts are simple examples of tight cuts. The following result is a characterization of tight cuts in matching covered bipartite graphs.
Proposition 2.1. Let G= (A, B) be a bipartite matching covered graph and C a cut of G. Then, C is tight if and only if there is a partition (A1, A2) of A, partition(B1, B2)ofB, where|B1|=|A1|+ 1and|A2|=|B2|+ 1, such that every edge ofC joins a vertex of B1 to a vertex of A2.
There are matching covered graphs which are free of nontrivial tight cuts: for example, K3,3, K4, C6 and the Petersen graph. Nonbipartite matching covered graphs free of nontrivial tight cuts are calledbricksand bipartite matching covered graphs free of nontrivial tight cuts are calledbraces.
2.2. Tight cut decomposition
Let G be a connected graph and C a cut of G. We shall refer to the graph obtained fromGby contracting a shore ofCto a single vertex as aC-contraction ofG. The following property is straightforward from the definition of tight cuts.
B
B1 B2
A
A1 A2
C
Figure 2. Tight cut in bipartite graph.
Proposition 2.2. Let Gbe a matching covered graph,C a tight cut ofG. Then, theC-contractionsG1 andG2 ofGare both matching covered. Moreover, for any setR⊆E(G), ifG1−RandG2−Rare both matching covered then G−R is also matching covered and C−R is a tight cut ofG−R.
Thus, given any matching covered graphG and a nontrivial tight cutC ofG, we can obtain two smaller matching covered graphsG1andG2which are the two C-contractions of G. If either G1 or G2 has a nontrivial tight cut, we can take its cut-contractions, in the same manner as above, and obtain smaller matching covered graphs. Thus, by repeatedly applying cut-contractions with respect to tight cuts, we can obtain a list of graphs which do not have nontrivial tight cuts (bricks and braces). This procedure is known astight cut decomposition.
Theorem 2.3 (see [11]). Any two applications of the tight cut decomposition procedure on a matching covered graphGproduce the same list of bricks and braces, except possibly for the multiplicities of edges.
In particular, the numbers of bricks and braces resulting from a tight cut de- composition of a matching covered graphGis independent of the tight cut decom- position; we shall call these the numbers of bricks and braces of G respectively.
We shall letb(G) denote the number of bricks ofG, andp(G) denote the number of bricks whose underlying simple graphs are Petersen graphs. The numberb(G) plays an important role in this paper. We remark that in view of Proposition 2.1, ifGis a bipartite matching covered graph thenb(G) = 0.
The next result shows the importance of tight cuts in the study of Pfaffian orientations.
Theorem 2.4 (see [10]). Let G be a matching covered graph and C :=∂(X)a tight cut ofG. Then Gis Pfaffian if and only if the two C-contractions ofGare Pfaffian.
It follows that once we have characterized Pfaffian bricks and braces, we have characterized all Pfaffian graphs. Thus, in the study of the Pfaffian orientation problem, we may restrict our attention to bricks and braces. Another important
remark is that by Theorem 2.4 and Proposition 1.6, if a matching covered graphG hasp(G)>0 thenGis non-Pfaffian.
2.3. Minors
Recall that a subgraphH of a graphGis aconformalsubgraph ofGifG−V(H) has a perfect matching. The property of being a conformal subgraph is transitive, that is, ifG is a conformal subgraph ofG andG is a conformal subgraph of G thenG is also a conformal subgraph ofG. Observe that a graph is a conformal subgraph of itself. By Theorem 1.2, we immediately deduce that a graph G is Pfaffian if and only if every conformal subgraph ofGis Pfaffian.
Let G be a matching covered graph, let u and v be two adjacent vertices of degree two in G, and let w be the neighbor of v different from u. Then ∂(X), whereX:={u, v, w}, is a tight cut ofG. One of the∂(X)-contractions is a brace on four vertices; letH denote the other∂(X)-contraction. Since a brace on four vertices is Pfaffian, it follows from Theorem 2.4 thatGis Pfaffian if and only ifH is Pfaffian. A graphG is aneven subdivision of a graph H ifGcan be obtained from H by even subdivisions of its edges. It follows from the above observation that a graph is Pfaffian if and only if each of its even subdivisions is Pfaffian.
A graphH is aconformal minor(or simply aminor) ofGif there exists an even subdivision ofH that is a conformal subgraph ofG. From the above observations one can deduce directly that:
Proposition 2.5. A graphGis Pfaffian if and only if every minor ofGis Pfaffian.
Little [9] proved that a bipartite matching covered graph is Pfaffian if and only if it does not containK3,3as a minor. In Section 4 we present an alternative proof of this result.
2.4. Ear decompositions
LetG be a connected graph. A single ear of G is a path P of odd length in G whose internal vertices (if any) have degree two inG. If P is a single ear of G then we denote by G−P the graph obtained from G by deleting the edges and internal vertices of P. The following theorem provides a decomposition of bipartite matching covered graphs.
Theorem 2.6(see[12], Th. 4.1.6). Given any bipartite matching covered graphG, there exists a sequence
G1⊂G2⊂. . .⊂Gr=G
of conformal matching covered subgraphs of G where (i) G1 = K2, and (ii) for 2≤i≤r,Gi−1=Gi−Ri, whereRi is a single ear ofGi.
Such decompositions do not exist for non-bipartite matching covered graphs.
For example,K4andC6 have no ear decomposition as in Theorem 2.6. However, every matching covered graph has an ear decomposition with a slight relaxation
of the above definition. To describe that decomposition, we require the notion of a double ear.
Adouble ear of Gis a pair (R1, R2), whereR1 andR2 are two vertex-disjoint single ears ofG. Anear of Gis either a single ear or a double ear ofG. IfR is an ear ofGthen we denote by G−Rthe graph obtained fromGby deleting the edges and internal vertices of the constituent paths ofR.
A single earR ofG is removable if the graphG−R is matching covered. A removable single ear of length one isa removable edge. A double earR= (R1, R2) ofGisremovableifG−Ris matching covered. In this case, it is to be understood that neitherR1 norR2 is a removable single ear. Aremovable ear ofGis either a single or a double ear which is removable. A removable double ear in which both constituent paths have length one is aremovable doubleton. Observe that a removable ear of a brick is either a singleton or a doubleton.
Anear decompositionof a matching covered graphGis a sequence G1⊂G2⊂. . .⊂Gr=G
of subgraphs of Gwhere (i) G1 =K2, and (ii) for 2 ≤i ≤ r, Gi−1 =Gi−Ri, whereRi is a removable ear ofGi. Note that each term of an ear decomposition ofGis a conformal matching covered subgraph ofG. The following fundamental theorem was established by Lov´asz and Plummer.
Theorem 2.7(the two-ear theorem [12], Th. 5.4.2).Every matching covered graph has an ear decomposition.
A removable earRof a matching covered graphGisb-removableif either (i)R is a single ear andb(G−R) =b(G) or (ii)Ris a double ear andb(G−R) =b(G)−1.
An ear decomposition of a matching covered graph is optimal if it uses the least possible number of double ears. In [5], Theorem 1.3, we proved that in every matching covered graphGthe optimal number of double ears isb(G) +p(G). For this we used a result [5], Theorem 6.2, that implies the following statement:
Theorem 2.8. Every matching covered graph distinct from K2 and such that p(G) = 0contains a b-removable ear.
3. Number of dissimilar Pfaffian orientations
We say that two orientationsD and D of a graphGaresimilar if there is a cutC such thatD=DrevC. We start with a very simple observation.
Proposition 3.1. Similarity is an equivalence relation on the set of all orienta- tions of a graph.
3.1. A lower bound on the number of dissimilar Pfaffian orientations of matching covered graphs
LetGbe a matching covered graph and letDbe a Pfaffian orientation ofG. We shall proceed to describe how to obtain 2b(G) Pfaffian reorientations ofD, no two
of which are similar. For this purpose, we consider a tight cut decomposition G of G. Given any subset B of bricks in G, we shall see that the directed graph obtained by reversing the orientations of all the arcs in R :=⊕B∈BE(B) is also a Pfaffian orientation ofG. The difficult part is to show that distinct subsets of bricks give rise to dissimilar Pfaffian orientations. The following lemma will be used in establishing that fact.
Lemma 3.2. Let G be a matching covered graph, G a tight cut decomposition ofG,B a subset ofG,R:=⊕B∈BE(B). The following properties hold:
(i) for each conformal circuit QofG,|R∩E(Q)|is even;
(ii) setR is a cut if and only if Bconsists solely of braces.
Proof. By induction on|V(G)|. IfBis empty then the assertion holds immediately.
We may thus assume thatBcontains at least one brick or one brace.
Consider first the case in whichGis either a brick or a brace, in which case G is the only element of B. Then,R =E(G). Every conformal circuitQ ofGhas even length, whence|R∩E(Q)|is even. By Proposition 1.3, setRis a cut if and only if|R∩E(Q)|is even, for every circuitQofG. That is,Ris a cut ofGif and only ifGis bipartite. The assertion holds ifGis a brick or a brace.
We may thus assume thatG has nontrivial tight cuts. LetC be a nontrivial tight cut of G that is used to produce G. Let G1 and G2 denote the two C- contractions ofG. Fori= 1,2, letGi and Bi denote the restrictions ofG and B, respectively, to bricks and braces ofGi, let Ri :=⊕B∈BiE(B). Then,{G1,G2} is a partition ofG,{B1,B2} is a partition ofB,Gi is a tight cut decomposition ofGi andBi⊆ Gi, fori= 1,2. Moreover,R=R1⊕R2.
To prove part (i) of the assertion, letQbe a conformal circuit ofG. Consider first the case in which E(Q) and C are disjoint. As C is tight in G, it follows thatQis a conformal circuit of one ofG1andG2. Adjust notation so thatQis a conformal circuit ofG1. Then,R∩E(Q) =R1∩E(Q). By induction hypothesis,
|R1∩E(Q)| is even. Thus,|R∩E(Q)| is even. Part (i) of the assertion holds in this case.
Consider next the case in which Qhas at least one edge in C. As C is tight in G, circuit Qcontains precisely two edges inC. Fori= 1,2, letQi denote the circuit ofGi spanned byE(Q)∩E(Gi). Then,Qiis a conformal circuit ofGi. By induction hypothesis,|Ri∩E(Qi)|is even. Thus,
R∩E(Q) = (R1⊕R2)∩E(Q) = (R1∩E(Q1))⊕(R2∩E(Q2)).
By induction hypothesis, each of |R1∩E(Q1)| and |R2∩E(Q2)| is even, thus
|R∩E(Q)|is also even. The proof of (i) is complete.
To prove part (ii) of the assertion, consider first the case in whichB consists solely of braces. Then, each graph inB1 is a brace. By induction hypothesis,R1
is a cut ofG1. Thus, R1 is a cut ofG. LetX1 be a shore of R1 in G. Likewise, R2is a cut ofG, letX2be a shore ofR2inG. Then,R=R1⊕R2=∂G(X1⊕X2) is a cut ofG. Part (ii) of the assertion holds if each graph inBis a brace.
Consider last the case in whichBcontains at least one brick. Then, at least one ofB1 andB2 contains at least one brick. Adjust notation so that B1 contains at least one brick. By induction hypothesis,R1is not a cut ofG1. By Proposition 1.3, G1 contains a circuit,Q1, such that|R1∩E(Q1)|is odd. If Q1 contains no edges inC thenQ1 is a circuit ofGandR∩E(Q1) =R1∩E(Q1), whence|R∩E(Q1)| is odd. By Proposition 1.3, setR is not a cut ofG.
We may thus assume that Q1 has (precisely two) edges in C, say e and f. LetMe and Mf denote two perfect matchings ofG that contain edgese and f, respectively. AsCis tight, edgeedoes not lie inMfandfdoes not lie inMe. Thus, theMe, Mf-alternating circuit S of G that contains edgee contains also edgef and no other edge ofC. The setE(G2)∩E(S) spans inG2 a circuit, sayQ2, that is conformal inG2. Moreover,E(Q1)∪E(Q2) spans in Ga circuit,Q. Then,
R∩E(Q) = (R1⊕R2)∩E(Q) = (R1∩E(Q1))⊕(R2∩E(Q2)).
Note thatR2∩E(Q2) =R2∩E(S). By part (i), withB2playing the role ofB, and sinceSis a conformal circuit ofG, it follows that|R2∩E(Q2)|is even. By defnition ofQ1,|R1∩E(Q1)|is odd. It follows that|R∩E(Q)|is odd. By Proposition 1.3, setRis not a cut ofG. The proof of (ii) completes the proof of the Lemma.
Corollary 3.3. Every Pfaffian matching covered graph Ghas at least 2b(G) dis- similar Pfaffian orientations.
Proof. By hypothesis,Ghas a Pfaffian orientation, sayD. LetG0be the collection of (preciselyb(G)) bricks of a tight cut decomposition ofG. For each subsetBofG0, letR(B) :=⊕B∈BE(B), letD(B) :=DrevR(B). LetD:={D(B) :B⊆ G0}.
By part (i) of the lemma and Theorem 1.2, the 2b(G) orientations in D are Pfaffian. To complete the proof, we now show that they are dissimilar. For this, letB1 andB2 denote two distinct subsets ofG0. Then, B:=B1⊕ B2 is a nonnull subset ofG0. Moreover,R(B) =R(B1)⊕R(B2), whenceD(B2) =D(B1) revR(B).
By the Lemma, part (ii),R(B) is not a cut ofG. Therefore,D(B1) andD(B2) are dissimilar. This conclusion is valid for any two distinct subsetsB1 andB2 ofG0.
The assertion holds.
3.2. Pfaffian extensions
In this section we use Theorem 2.8 in order to prove that every Pfaffian matching covered graphGhas precisely 2b(G)dissimilar Pfaffian orientations. We also show how to determine Pfaffian orientations for Pfaffian matching covered graphs.
Theorem 3.4. Let G be a matching covered graph, D a collection of 2b(G) dis- similar Pfaffian orientations of G. Let R be a b-removable ear of G, Q a con- formal circuit of G that contains some edge of R, D0 an orientation of G such that (i)D0−Ris a Pfaffian orientation ofG−Rand (ii)Qhas odd parity inD0. IfG−Rhas precisely 2b(G−R) dissimilar Pfaffian orientations thenD0 is similar to some orientation inD.
Proof. Let G := G−R, t := 2b(G). Denote by D1, D2, . . . , Dt the t dissimilar Pfaffian orientations inD. Fori= 0,1, . . . , t, letDi:=Di−R. Fori >0,Di is a Pfaffian orientation ofG, becauseDi is a Pfaffian orientation ofG. Moreover, D0 is a Pfaffian orientation ofG, by hypothesis.
Consider first the case in whichR is a single ear. In that case,b(G) =b(G).
By hypothesis,Ghas preciselytdissimilar Pfaffian orientations. Therefore, there are indicesiand j such thatDi andDj are similar, where 0≤i < j≤t. LetC denote a cut ofG such thatDi =DjrevC. Let X denote a shore ofC in G. LetC:=∂G(X). Then,Dj :=DjrevC is similar to Dj, whenceDj is a Pfaffian orientation ofG. Moreover, for each edge ofG, its orientations in Di and in Dj coincide. Ifi = 0 then Q has odd parity in Di, by hypothesis. If i > 0 then Q has odd parity inDi, becauseDiis Pfaffian. As Dj is Pfaffian,Qhas odd parity inDj. As R is a single ear ofGandQcontains an edge ofR, it contains all the edges ofR. Each edge ofE(Q)−E(R) has the same orientation inDi and inDj. We conclude thatR has the same parity inDiand inDj. By Proposition 1.5,Di
and Dj are similar. Thus, Di and Dj are similar. As the orientations in D are dissimilar, it follows thati= 0. The assertion holds ifR is a single ear.
Consider next the case in whichRis a double ear. In that case,b(G) =b(G)−1.
By hypothesis, G has precisely t/2 dissimilar Pfaffian orientations. Therefore, there are indicesi,j and ksuch thatDi,Dj and Dk are similar, where 0 ≤i <
j < k≤t. LetCj andCk denote cuts ofGsuch thatDi=DjrevCj =Dk revCk. LetXj andXk denote shores ofCj andCk inG, respectively. LetCj :=∂G(Xj), Ck :=∂G(Xk). Then, Dj :=DjrevCj is similar toDj, whence Dj is a Pfaffian orientation of G. Likewise, Dk := DkrevCk is similar to Dk, whence Dk is a Pfaffian orientation ofG. Moreover, for each edge of G, its orientations in Di, Dj and in Dk coincide. If i= 0 then Qhas odd parity inDi, by hypothesis. If i >0 thenQhas odd parity inDi, becauseDiis Pfaffian. AsDj andDk are both Pfaffian, Q has odd parity in both. Let R1 and R2 denote the two constituent paths ofR. Adjust notation so thatQcontains an edge ofR1. Then, Qcontains all the edges ofR1. IfQdoes not contain any edge inR2thenG−R2is matching covered, whenceR2 is a removable ear of G, a contradiction to the definition of removable double ear. We conclude that Q has edges inR2, whence it contains all the edges of R2. It follows that Q contains all the edges ofR. Path R1 has the same parity in at least two ofDi,Dj andDk. As each edge of E(Q)−E(R) has the same orientation in those three orientations, and sinceQhas odd parity in those three orientations, it follows that in at least two of the orientationsDi, Dj andDk, each ofR1and R2 has the same parity. By Proposition 1.5, at least two ofDi, Dj and Dk are similar. Thus, at least two ofDi, Dj and Dk are similar.
As the orientations inDare dissimilar, it follows thati= 0 andDi is similar to (precisely) one ofDj andDk. The assertion holds ifR is a double ear.
Corollary 3.5. Every Pfaffian matching covered graph Ghas precisely2b(G)dis- similar Pfaffian orientations.
Proof. By induction on|E(G)|. Lett:= 2b(G). By hypothesis,Gis Pfaffian. By Corollary 3.3,Ghas a collectionDof tdissimilar Pfaffian orientations. To prove
the assertion, we now show that every Pfaffian orientationD of G is similar to some orientation inD.
The assertion holds immediately if G is K2. We may thus assume that G is distinct fromK2. AsG is Pfaffian, we have thatp(G) = 0. By Theorem 2.8, G has ab-removable earR. AsDis Pfaffian,D−Ris a Pfaffian orientation ofG−R. Every conformal circuit ofGhas odd parity inD. By induction hypothesis,G−R has precisely 2b(G−R)dissimilar Pfaffian orientations. By Theorem 3.4,Dis similar to some orientation in D. This conclusion holds for each Pfaffian orientationD
ofG.
Corollary 3.6. Let G be a matching covered graph, R a b-removable ear of G, D an orientation ofG,Qa conformal circuit ofGthat contains some edge ofR. Assume that (i)D−Ris a Pfaffian orientation ofG−Rand (ii)Qhas odd parity inD. Then,Gis Pfaffian if and only ifD is a Pfaffian orientation of G.
Proof. IfDis a Pfaffian orientation ofGthenGis certainly Pfaffian. To prove the converse, assume thatGis Pfaffian. By Corollary 3.3,Ghas a collectionDof 2b(G) dissimilar Pfaffian orientations. By Corollary 3.5, G−R has precisely 2b(G−R) dissimilar Pfaffian orientations. By Theorem 3.4,Dis similar to some orientation inD. Thus,D is a Pfaffian orientation ofG.
3.3. Polynomial reducibilities
In this section we give a polynomial algorithm that, given a matching covered graph G, produces an orientation D of G such that G is Pfaffian if and only ifD is Pfaffian. We also give polynomial reductions relating two problems: (i) to determine whether or not a given matching covered graph is Pfaffian and (ii) to de- termine whether or not a given orientation of a matching covered graph is Pfaffian.
We need to record first some facts concerning the complexity of some problems related to matching covered graphs.
The literature has several polynomial algorithms to determine, given a graphG, a perfect matching ofG, or alternatively, a certificate thatGhas no perfect match- ings; the certificate is a setSof vertices ofGsuch that the number of odd compo- nents ofG−S is strictly larger than|S|. Using any such algorithm, it is possible to determine in polynomial time whether or not a given graphGis matching cov- ered. In linear time it is possible to verify whether or notG is connected. For each edgeeofG, if uandv denote the ends ofe, Ghas a perfect matching that contains edgeeif and only if graphG−u−v has a perfect matching.
We remark that to determine, in polynomial time, a conformal circuit of a given matching covered graphGdistinct fromK2 that contains a given edgeeofG, it suffices to determine a perfect matchingM ofGthat contains edgeeand a perfect matchingNofGthat does not contain edgee, and then take theM, N-alternating circuit ofGthat contains edgee.
To determine the set of removable ears of a matching covered graphGin poly- nomial time, letP denote the set of pathsP of Gof maximum length such that each internal vertex ofP has degree two. If Gis not a circuit, thenP induces a
partition ofE(G) andP may be determined in linear time. The set of removable ears of an even circuit is trivially determined, so assume thatG is not a circuit.
Every constituent path of a removable ear ofGlies inP and has odd length. Thus, the set of removable single ears ofGis the set of those pathsRinPthat have odd length and such thatG−R is matching covered. Likewise, the set of removable double ears of Gis the set of those vertex-disjoint pairs {R1, R2} of paths in P such that each ofR1andR2has odd length,G−R1−R2is matching covered and neitherR1 norR2 is removable.
We now make some remarks concerning the determination, in polynomial time, of the numberb(G) of bricks of any tight cut decomposition ofG. In linear time, determine whether or notGis bipartite. If Gis bipartite thenb(G) = 0. Assume thus thatGis nonbipartite. For each pair{u, v} of distinct verticesuandvofG, determine whether or notG−u−v is connected and has a perfect matching. If, for some such pair,G−u−v has no perfect matching, then letB :=S∪ {u, v}, where S is a certificate of the nonexistence of a perfect matching in G−u−v.
AsGis matching covered, it follows that the number of odd components ofG−B is equal to|B|, no edge ofG has both ends in B and each component ofG−B is odd. As G is nonbipartite, at least one (odd) component of G−B, say K, is nontrivial. In that case, C := ∂(V(K)) is a nontrivial tight cut of G. Then, b(G) = b(G1) +b(G2), where G1 and G2 are the two C-contractions of G, the algorithm procedes recursively. Assume thus thatG−u−vhas a perfect matching, for each pair{u, v} of distinct vertices ofG. Then,Gis said to be bicritical. We thus have a polynomial algorithm to determine whether or not a matching covered graph is bicritical.
We may assume that Gis bicritical. If, for some pair{u, v},G−u−v is not connected, then, as G is bicritical, each component of G−u−v is even. For any component K of G−u−v, C := ∂(V(K)∪ {u}) is a nontrivial tight cut of G. Again, recursively determine b(G1) and b(G2), where G1 and G2 are the C-contractions ofGand determineb(G) =b(G1) +b(G2).
Finally, if Gis bicritical and 3-connected, then Gis a brick, by the following assertion, proved by Edmonds, Lov´asz and Pulleyblank [6].
Theorem 3.7. A nonbipartite matching covered graph Gis a brick if and only if it is 3-connected and bicritical.
In that case, b(G) = 1. We have thus very briefly described a polynomial algorithm for determiningb(G) for a given matching covered graphG.
We end these preliminary remarks with the following observation. In polynomial time, we know how to determine the set of removable ears ofGand also, for each such earR, the numbers b(G) and b(G−R). Therefore, in polynomial time we know how to determine the set ofb-removable ears ofG.
Lemma 3.8. There is a polynomial algorithm that, given a matching covered graph G, either determines (i) a conformal subgraphH ofGsuch that p(H)>0, or (ii) an ear decomposition
G1⊂G2⊂. . .⊂Gr=G
ofGwhere for 2≤i≤r,Gi−1=Gi−Ri, andRi is ab-removable ear of Gi.
Proof. IfGisK2thenr:= 1 andG1:=G. Assume thus thatGis notK2. Deter- mine, in polynomial time, ab-removable ear ofG, if one exists. By Theorem 2.8, if no such ear exists then p(G) > 0 and we are done. If a b-removable ear R is found, then recursively run the algorithm for matching covered graphG−R. Any conformal subgraph ofG−R is a conformal subgraph ofG. IfG−R has a conformal subgraphH such thatp(H)>0, thenH is a conformal subgraph ofG and we are done. Alternatively, an ear decomposition
G1⊂G2⊂. . .⊂Gr−1=G−R
is obtained, where for eachi, 2≤i < r, Gi−1=Gi−Ri, andRi is ab-removable ear of Gi. Add to that decomposition graph Gr := G as its last term, where Rr:=Ris theb-removable ear ofGrsuch thatGr−1=Gr−Rr. Theorem 3.9. There exists a polynomial algorithm that, given a matching covered graph G, determines an orientation D of G such that G is Pfaffian if and only ifD is a Pfaffian orientation ofG.
Proof. Run the algorithm described in the proof of Lemma 3.8. We now define the orientationD as follows. If a conformal subgraphH ofGsuch thatp(H)>0 has been obtained, thenGis not Pfaffian, and any orientationDofGsatisfies the assertion. We may thus assume that an ear decomposition
G1⊂G2⊂. . .⊂Gr=G
was determined, where, fori= 2,3, . . . , r,Gi−1=Gi−Ri, andRiis ab-removable ear ofG. Orient G1 arbitrarily. Fori = 2,3, . . . , r, extend the orientation Di−1
of Gi−1 to an orientation Di of Gi as follows: determine in polynomial time a conformal circuitQi ofGi that contains some edge inRi. Orient the edges ofRi
so thatQi has odd parity inDi.
We now show, by induction oni, thatGiis Pfaffian if and only ifDiis a Pfaffian orientation ofGi. This assertion holds immediately for i= 1. Assume thus that i >1. IfDi is a Pfaffian orientation ofGi, thenGi is certainly Pfaffian. To prove the converse, assume that Gi is Pfaffian. Then, Gi−1 is Pfaffian. By induction, Di−1 is a Pfaffian orientation of Gi−1. By Corollary 3.6, it follows thatDi is a Pfaffian orientation ofGi. In particular,Gis Pfaffian if and only ifD:=Dr is a
Pfaffian orientation ofG.
With the next two corollaries we establish the result first proved by Vazirani and Yanakakis [16], that it is no easier to decide whether a given orientation of a graph is Pfaffian than deciding that the graph is Pfaffian and finding a Pfaffian orientation if it is.
Corollary 3.10. The problem of determining whether or not a given match- ing covered graph is Pfaffian is polynomially reducible to the problem of deciding whether or not a given orientation of a matching covered graph is Pfaffian.
Corollary 3.11. The problem of determining whether or not a given orientationD of a matching covered graphGis Pfaffian is polynomially reducible to the problem of deciding whether or not a given matching covered graph is Pfaffian.
Proof. Determine whether Gis Pfaffian. If Gis not Pfaffian then certainlyD is not Pfaffian. Assume thus thatGis Pfaffian. Then,p(H) = 0, for each conformal subgraph H of G. Run the algorithm described in the proof of Lemma 3.8, in order to obtain an ear decomposition
G1⊂G2⊂. . .⊂Gr=G
ofG, where, fori= 2,3, . . . , r,Gi−1=Gi−Ri, andRi is ab-removable ear ofG. For eachi, 1≤i≤r, letDi denote the restriction ofDto Gi.
For i = 1,2, . . . , r, we now determine, in polynomial time, whether Di is Pfaffian, and, if the answer is negative, a conformal circuit Q of Gi that has even parity inGi. Ifi= 1, thenDi is certainly Pfaffian. Assume thus thati >1.
If Di−1 is non-Pfaffian, then letQ be a conformal circuit of Gi−1 that has even parity inDi−1. Then, Qis a conformal circuit of Gi that has even parity in Di. Assume thus that Di−1 is Pfaffian. In polynomial time, determine a conformal circuit ofGi that contains some edge in the b-removable earRi of Gi. If Qi has even parity inDi thenDiis certainly non-Pfaffian. Assume thus thatQihas odd parity inDi. By Corollary 3.6, it follows thatDi is a Pfaffian orientation ofGi. In particular, the algorithm either determines thatD is Pfaffian, or alternatively, returns a conformal circuit ofGthat has even parity inD.
4. A Characterization of Pfaffian bipartite graphs
The first results in this section provide the basis for the characterization of Pfaffian bipartite graphs. Recall that braces are bipartite matching covered graphs free of nontrivial tight cuts. Braces have an important property concerning remov- able edges.
Lemma 4.1 (see [2], Lem. 3.2). Let Gbe a brace on at least six vertices. Then every edge ofGis removable.
The following lemma provides a characterization of non-removable edges in matching covered bipartite graphs.
Lemma 4.2 (see [12], Th. 4.1.1). Let G[A, B] be a bipartite matching covered graph distinct fromK2and letebe an edge ofG. Then, eis not removable inGif and only if there is a partition{A, A} ofA and a partition {B, B} ofB with
|A|=|B|such that eis the only edge joining a vertex inA to a vertex inB. Many properties that do not hold in general for bipartite matching covered graphs hold for bipartite graphs obtained from the deletion of a removable edge from a brace. We now describe some of those properties.
LetGbe a matching covered graphGwith bipartition{A, B}. For each shoreX of a tight cut C, the numbers |X ∩A| and |X ∩B| differ by exactly one, by
Proposition 2.1. Whichever of X ∩A and X∩B is the largest, it is called the majority partofX, whereas the other is theminority partofX. The next property is immediate from the fact that braces are free of tight cuts and Proposition 2.1.
Proposition 4.3. Let G be a brace, e a removable edge of G,C :=∂(X)−ea nontrivial tight cut ofG−e. Then,eis the only edge ofGthat has its ends in the minority parts ofX and ofX.
Lemma 4.4. LetGbe a brace,ea removable edge ofG,C:=∂(X)−ea nontrivial tight cut ofG−e. Then, each edge ofC is removable inG−e.
Proof. Assume to the contrary that there is an edgee inC that is not removable inG−e. Then,e is not removable in some (bipartite)C-contraction ofG−e, by Proposition 2.2. Adjust notation so thateis not removable in theC-contractionH obtained from G by contractingX to a single vertex x. Let {A, B} denote the bipartition ofH. Adjust notation so thatxlies inA.
Lety denote the end of e in X. Thus,y lies in the majority part of X. By Lemma 4.2, there exists a partition (B, B) ofB and a partition (A, A) of A such that|A|=|B| ande is the only edge of H that joins a vertex of A to a vertex of B. Then, Y := A∪B∪ {y} is the shore of a nontrivial tight cut of G−e. Moreover, edgeehas one end in X, the other in X∩A, the minority part ofX. Thus, no end of e lies in B. We conclude that Y is the shore of a nontrivial tight cut ofG, a contradiction. As asserted, every edge ofCis removable
inG−e.
Suppose thatGis a brace andeis a removable edge ofG. It turns out that one can always find at least two removable edges ofG−e incident with any vertex, provided that vertex is of degree greater than two inG−e, as shown in the next result.
Lemma 4.5. Let G[A, B]be a brace, ea removable edge ofG,v any vertex ofG.
If the degree of v is greater than two in G−e then at most one edge of G−e incident withv is not removable inG−e.
Proof. If every edge ofG−eincident withvis removable inG−ethen the assertion holds immediately. We may thus assume that there is an edgee incident with v that is not removable inG−e. We must now show that every edge of∂(v)−e−e is removable inG−e.
By Lemma 4.2, there is a partition {A, A} of A and a partition {B, B} of B with |A| = |B| such that e is the only edge of G−e joining a vertex in A to a vertex in B. Assume, without lost of generality, thatv ∈A. Then X := (A−v)∪B is the shore of a (possibly trivial) tight cutC ofG−e.
Consider first the case in whichX is a singleton. In that case, each edge of
∂(v)−e−e joins v to the only vertex ofX. By hypothesis, the degree ofv in G−e is three or more. Therefore, the edges of∂(v)−e−e are multiple edges.
We conclude that the assertion holds in this case.
We may thus assume thatX is not a singleton. AsXcontains all the vertices ofBand also vertexv, it follows thatX is not a singleton either. Thus,C is a
