Annales de l’institut Fourier (AIF)

Gavrilov, Lubomir

Abelian integrals related to Morse polynomials and perturbations of plane hamiltonian vector fields

Bouchet, André

ANNALES DE L’INSTITUT FOURIER
ANDRÉ BOUCHET
Bipartite graphs that are not circle graphs
Annales de l’institut Fourier, tome 49, no 3 (1999), p. 809-814
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Ann. Inst. Fourier, Grenoble
49, 3 (1999), 809-814
BIPARTITE GRAPHS THAT ARE NOT
CIRCLE GRAPHS
by Andre BOUCHET
The graphs considered in this note are finite and simple. Let G be a
graph defined on the vertex-set V, with edge-set E. An edge with ends x
and y is denoted by xy. The complement of G is the graph G, defined on
the same vertex-set as G^ such that xy is an edge of G if and only if xy is
not an edge of G. We denote by E the edge-set of G.
Set n == \V\. Naji [9] has defined a linear system of equations v(G)
with n(n — 1) unknowns a^y in GF(2), defined for every ordered pair (.r, y)
of distinct vertices of G. The equations are of two types:
T(x, y , z) : a^y + a^z + o-yz + a^y = 1
if either (i) xy and .r^ are edges of G and ̂  is an edge of G or (ii) xy and
rcz are edges of G and ?/^ is an edge of G,
E(y, z) : a.yz + a^y = 1
if ?/^ is an edge of G.
Originally the first subsystem of equations was defined only in Case
(i). In Case (ii) the equation T(x,y,z) was replaced by a^y + o-xz == 0-
The two systems of equations are seen to be equivalent by making a linear
combination of T(x^ y^ z) and E(y^ z).
A circle graph is an intersection graph of finitely many chords of a
circle. The following characterization of circle graphs is part ofNaji's thesis.
The original proof, which is long and difficult, has not been published. A
Keywords: Circle graph — Binary matroid.
Math. classification: 05C - 05C75.
810 ANDRE BOUCHET
short proof has been recently derived by Gasse [5] from a characterization
of circle graphs by three obstructions, due to the author [3].
THEOREM 1. — A graph G is a circle graph if and only ifz/(G) has
a solution.
Here we focus on the reduced system of equations ^(G), which
only contains the equations T(x,y,z). If ^(G) and v{G) were equivalent
systems, then the complement of any circle graph would be a circle graph.
However, it is easily checked that the 6-circuit is a circle graph, and that its
complement is not. This example also shows that the converse of Corollary 3
fails.
THEOREM 2. — If yp(G) has a solution for a bipartite graph G,
then v(G) also has a solution.
Here we mention an interesting application to bipartite circle graphs.
It has been proved by de Fraysseix [4] that a bipartite graph is a circle
graph if and only if it is the fundamental graph of a planar graph. One of
the nice corollaries of Naji's characterization is that it provides algebraic
characterization of planarity. If you want to check whether a graph is
planar you can construct Naji's system for one of its fundamental graphs.
Theorem 2 is interesting as it removes some redundant constraints from
the system.
COROLLARY 3. — If a bipartite graph G is not a circle graph,
then G also is not a circle graph.
We first recall some definitions and results used in the sequel. The
symmetric difference of two subsets of edges P and Q is PAQ := (P \ Q) U
(Q \ P). If X is a subset of vertices of G, then we denote by V(X) the set
of edges of G having one end in X and one end in V \ X. The set VX is a
cut. If a; is a vertex then we set V(:r) = V({a;}), and we call V(a;) a vertex-
cut. Every cut can be expressed as a symmetric difference of vertex-cuts.
A circuit of G is a subset C of edges such that the subgraph of G induced
by G, denoted by G[G], is connected and regular of degree 2. The circuit
is chordless if the edge-set of the subgraph G[V(G)] (induced by V(C), the
vertex-set of G) is equal to G. It is known that a subset F of edges is a cut
if and only if |mG| is even, for every circuit G. It is also known that every
circuit of a graph can be expressed as a symmetric difference of chordless
circuits. (We refer the reader to [10] for a proof.) By combining the two
BIPARTITE GRAPHS THAT ARE NOT CIRCLE GRAPHS 811
preceding properties we obtain the following one.
PROPOSITION 4. — A subset F of edges of a graph is a cut if and
only if |r D C\ is even for every chordless circuit.
If a = (cKxy : ^5 V € V, x 7^ y ) is a solution of ^(G), then we denote
by Ea the subset of edges xy of (7 such that a^y + o;i/a; = 1-
LEMMA 5. — Let G be a graph and let a be a solution ofVR(G).
IfC is a chordless even circuit ofG, then \Ea D C\ is even.
Proof. — Let V Q ^ V I ^ ' ' ' ' > V 2 i - i be the successive vertices of C. For
each triple {a;, ?/, z} giving rise to an equation T(x^ y , z) set r(x^ y , z) =
Oixy + 0(-xz + Q-yz + O^zy Set
21-1 21-1
x =^ T(^_i,^,-^+i) 4- ̂  ^ T(^,^-,^4-i),
1=0 i=o jv{i-i,^+i}
where the arithmetic on the indices is mod. 21. The reader will verify that
21-1
x = ̂  (^z^+i + o'^+ivj?
1=0
which implies
(1) X=\EaC}C\ (mod 2).
The number of terms of the form r(vp,Vq^Vr) in the expression of X is
even. Since each of these terms has value 1 in ^(G), it follows
(2) X = 0 (mod 2).
The equalities (1) and (2) imply \E^ H C\ == 0 (mod 2). D
LEMMA 6. — Let a be a solution of vn(G) and let Y be a cut of
G. There exists a solution a' ofi^p{G) such that
E^ = E^r.
Proof. — Since F is a symmetric difference of vertex-cuts, it is
sufficient to prove the lemma when F = V(v), for some vertex v. Change
Oivx into 1 -I- OL^, for every vertex x different from v, and change a^v into
1 4- o^xvj for every vertex x joined to v by an edge of G. We easily check
that we obtain a new solution a' of vp^(G). For every edge xy, the values
812 ANDRE BOUCHET
<^xy + o^yx and a' xy + o/^a; are distinct if and only if e belongs to V(v). It
follows Ec,' = £^AVv. D
Proof of Theorem 2. — Let a be a solution of vp{G). Lemma 5 and
Proposition 4 imply that Ea is a cut of G. Since G is bipartite the edge-set
of G is also a cut. Therefore F = E/^Ea is a cut. Lemma 6 implies the
existence of a solution a' of VR(G) such that Eo,i = Ea/^T = E. Hence
every edge xy of G satisfies a' xy + oi! yx = 1, and a' is a solution of v{G).
D
Proof of Corollary 3. — Assume indirectly that G is a circle graph.
Then v(G) has a solution by Naji's theorem. Hence I^R^G) has a solution.
The systems yp(G) and ^p{G) are equal. Hence yp(G) has a solution.
Theorem 2 implies that v(G) has a solution. NajFs theorem implies that
G is a circle graph, a contradiction. D
QUESTION. — Naji's result is difficult to prove. Is there an elemen-
tary direct proof of Corollary 3?
The influence of Frangois Jaeger
in the study of circle graphs.
In the 70's the main problem was to find a good characterization
of circle graphs, which was expected by many researchers, and to devise
a recognition algorithm of polynomial complexity. An approach was to
consider the binary matroids generated by the neighborhoods of a circle
graph.
If A is a collection of subsets of a finite set E, then the collection
of minimal nonempty subsets of E that can be expressed as symmetric
differences of subsets in A is the collection of circuits of a binary matroid,
which we call the binary matroid generated by A. If a; is a vertex of a simple
graph G = (V,E), then N(x) = {y e V : xy € E} is the neighborhood
of x.
If G is a bipartite graph with classes X and V, then de Fraysseix
[4] proved that G is a circle graph if and only if the binary matroids
generated by {N(x) : x e X} and {N(x) : x € Y} are graphic (which
is equivalent to say that G is a fundamental graph of a planar graph).
BIPARTITE GRAPHS THAT ARE NOT CIRCLE GRAPHS 813
This is a good characterization of bipartite circle graphs because graphic
matroids are characterized by excluding a finite number of excluded minors,
by a theorem of Tutte [11].
A main result of Francois Jaeger [6], [7], [8], when G is a simple
graph, says that, if P is a subset of V, then the binary matroid generated
by {N(x) U P H {x} : x € V} is graphic. If G is a circle graph and x is
any vertex of G, then the graph obtained from G by replacing the induced
subgraph G[N(x)] by the complementary subgraph is also a circle graph. A
graph derived from G by performing a sequence of such transformations is
said to be locally equivalent to G. If we apply Jaeger's result to the graphs
locally equivalent to G, one proves that G is a circle graph if and only if,
for every pair of subsets P and Q of V, the binary matroid generated by
{Q 0 (N(x) U P D {x}) : x € V} is graphic [1]. Unfortunately the converse
does not hold; the 5-wheel is not a circle graph and satisfies the preceding
condition. It would be interesting to compare the class of circle graphs with
the class of graphs satisfying Jaeger's conditions. A good characterization
of circle graphs can be found in [3].
Let A(G) be the adjacency matrix of a simple graph, with coefficients
in the field GF(2). Francois proved that, if A(G) has an inverse, then it is
the adjacency matrix of a graph G' locally equivalent to G. Accordingly G'
is a circle graph if G is a circle graph.
Francois was also the origin of fruitful ideas concerning circle graphs.
Naji prepared his thesis with him and Francois foreseen the importance of
circle graph orientations, which led to the equations studied in that note.
The matrix A(G) considered above has an interesting property, when G is
any circle graph; it can be transformed into an antisymmetric matrix with
integral coefficients, by changing into -1 some values equal to 1, in such
a way that every principal minor has a determinant equal to 0 or 1. This
property, which is now referred to principal unimodularity, is related to
Poincare duality. I gave a simple direct proof of the principal unimodularity
property by using an idea of Francois [2].
BIBLIOGRAPHY
[1] A. BOUCHET, Graphic presentations of isotropic systems, J. Combin. Theory, Ser.
B, 45 (1988), 58-76.
[2] A. BOUCHET, A characterization of unimodular orientations of simple graphs, J.
Combin. Theory, Ser. B, 56 (1992), 45-54.
814 ANDRE BOUCHET
[3] A. BOUCHET, Circle graph obstructions, J. Combin. Theory, Ser. B, 60 (1994),
107-144.
[4] H. de FRAYSSEIX, A characterization of circle graphs, Europ. J. Combin., 5 (1984),
223-238.
[5] E. GASSE, A short proof of a circle graph characterization, Discrete Math., 173
(1997), 277-283.
[6] F. JAEGER, Graphes de cordes et espaces graphiques, Europ. J. Combin., 4 (1983),
319-327.
[7] F. JAEGER, On some algebraic properties of graphs, in Progress in Graph Theory
(Waterloo, Ont. 1982), Academic Press, Toronto, Ont., 1984, 347-366.
[8] F. JAEGER, Graphic description of binary spaces, in Matroid Theory (Szeged,
1982), vol. 40 of Colloq. Math. Soc. Janos Bolyai, North-Holland, Amsterdam
(1985), 215-231.
[9] W. NAJI, Reconnaissance des graphes de cordes, Discrete Math., 54 (1985), 329-
337.
[10] K. TRUEMPER, Matroid Decomposition, Academic Press, 1992.
[11] W. T. TUTTE, Lectures on matroids, J. Res. Nat. Bur. Standards, Sect. B, 69b
(1965), 1-47.
Andre BOUCHET,
Universite du Maine
Departement d'lnformatique
72017 Le Mans Cedex (France).


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