AbstractA property of unimodularity is introduced for antisymmetric integral matrices. It is satisfied by the adjacency matrix of a circle graph provided with a Naji orientation [8]. In a further paper we shall interprete this result in terms of symmetric matroids introduced in [2]. In this communication we give a direct proof by means of techniques used in [1] for an algorithmic solution of the Gauss problem on self-intersecting surves in the plane

Bouchet, André

Elsevier - Publisher Connector 

Discrete Mathematics 66 (1987) 203-208 
North-Holland 
203 
COMMUNICATION 
UNIMODULARITY  AND C IRCLE  GRAPHS*  
Andr6 BOUCHET 
Centre de Recherche en Informatique et Combinatoire, Universit~ du Maine, 72017 Le Mans 
Cedex, France 
Received 3 March 1987 
Communicated by C. Benzaken 
A property of unimodularity is introduced for antisymmetric integral matrices. It is satisfied 
by the adjacency matrix of a circle graph provided with a Naji orientation [8]. In a further 
paper we shall interprete this result in terms of symmetric matroids introduced in [2]. In this 
communication we give a direct proof by means of techniques used in [1] for an algorithmic 
solution of the Gauss problem on self-intersecting curves in the plane. 
Let A = (A~w: v, w ~ V)  be an antisymmetric integral matrix. For each W ~ V 
we let A[W] = (A~w: v, we  W). We are interested in the following property of 
unimodularity: 
det (A[W])e{-1 ,  O, +1}, W~_ V. (o0 
Our graphs will be simple. An oriented graph is a simple graph where an initial 
end and a final end have been distinguished for every edge. The adjacency matrix 
of an oriented graph G is the antisymmetric (0, +l)-matrix A = (Aw: v, w e 
V(G) )  such that Aow = +1 if and only if vw is an edge oriented from v to w. The 
orientation of G is said to be unimodular if A satisfies Property (e 0. 
Let m be a word such that each letter occurring in m occurs precisely twice. 
We say that m is a double occurrence word. An  alternance of m is a non- 
ordered pai r  v 'v"  of distinct letters such that we meet alternatively 
• . .  v ' .  . . v" .  . . v ' .  . . v".  . . when reading m. The alternance graph G(m)  is the 
simple graph whose vertices are the letters of m and whose edges are the 
alternances of m. From a geometric point of view, alternance graphs can be 
interpreted as intersection graphs of chords of a circle, and they are more widely 
known as circle graphs. These graphs are also related to stack sorting techniques 
in a paper by Even and Itai [4]. The reader will find in [6] a survey of some results 
on circle graphs. Naji [8] found recently a good characterization f circle graphs 
by means of particular orientations. These are precisely these orientations which 
we will consider. 
* Partially supported by PRC Mathematique et Informatique. 
0012-365X/87/$3.50 ~) 1987, Elsevier Science Publishers B.V. (North-Holland) 
204 A. Bouchet 
e 
d~ < 
Fig. 1. 
b 
Let V be the set of letters of m, and let V +={v+:v•V} and V-= 
{v-: v • V} be disjoint copies of V. A separation of m is any word # over 
V + U V- obtained by replacing in m the two occurrences of each letter v by v + 
and v-. If v'v" is an altemance of m, and . . ,  v'S'.., v "~'. .. v ' -S ' . . ,  v " - " . . ,  is 
the succession i  # of the letters belonging to {v '+, v'- ,  v "÷, v"-}, then the edge 
v'v" of G(m) will be directed from v' to v" ifs'  =s", from v" to v' otherwise. Let 
D(#) be the oriented graph so defined, and let A(#) = (Avw: v, w • V) be the 
adjacency matrix of D(#). 
A rotation of a word xlx2.. .Xr is its transformation into any word 
xixi+l. • • x~xl. . ,  xi-~. We notice that G(m) and D(#) are invariant after rotations 
of m and #. Since each element of V + U V- occurs precisely once in #, we will 
identify # to a cyclic permutation of V+U V-. Where a~ is the involution over 
V + U V-  which exchanges each pair {v +, v-},  we let #* = # o a~, the composition 
of the permutations # and a~. If #* is a cyclic permutation, we define D(#*) and 
A(#*) like D(#) and A(#). To each orbit P of #* we attach the integral column 
matrix X(P) = (X," v • V) defined by X~ = + 1 if v + • P and v-  ~ P, X~ = -1  if 
v- • P and v ÷ ~ P, X,, = 0 otherwise. 
Example. For m = aebadecdbc and # = a+e+b-a-d+e-c-d-b+c+, Fig. 1 depicts 
D(#). We have #* = (a+d+b+a-e+c-)(c+d-e-b-). 
Property. An orbit P of #* satisfies X(P )= 0 if and only if #* is a cyclic 
permutation. 
Proof. X(P) = 0 if and only if any pair {v +, v-} which intersects P is included in 
P, which implies that P is a union of orbits of tr. Therefore P is also a union 
Unimodularity and circle graphs 205 
of orbits of the composition ft*o a~ = ft. The result follows because ?t is a cyclic 
permutation. [] 
"I'neorem. If  P is an orbit of #*, then A(#)X(P) = O. If  ft* is a cyclic permutation, 
then A(ft*) = A(#) -~. 
Proof. For each v • V, we suppose that v + is the column-matrix indexed over V 
whose components are null at the exception of the v-component equal to +1, and 
we let v -=-v  +. For each word ~' ~-PlP2...Ps which letters p~, P2, . . - ,  Ps in 
V+U V- ,  let H(p ' )= E (Pi: l<~i<~s). So we have H(#) =0.  Let us say that #'  
wraps above ~ ifp~+l follows p~ in # (after eventually rotating #) for 1 ~< i ~<s - 1. 
Moreover we say that ~'  is closed if P l follows p~ in #. Clearly 
(i) H(# ' )  = 0 if # '  is wrapped above # and is closed. 
For each v • V we denote by S(v ÷) and S(v-) the subwords of # such that, after 
eventually rotating #, ~ = v+S(v+)v-S(v-). 
(ii) A(ft)v + = -H(S(v+))  = H(S(v-))  = -A(#)v - ,  v • V. 
(iii) S(x)S(y) wraps above # if and only if x --- ft*(y). 
It is easy to verify (ii). To prove (iii) we notice that the letter which follows 
S(x) in ft is equal to a~(x), when the letter which precedes S(y) is y. Therefore 
S(x)S(y) wraps above ?t if and only if yot(x) is a pair of successive letters in ft, 
which means x = #* (y). 
Let (Xk...X2Xl) be the cyclic permutation induced over an orbit P of #*. 
Following (iii), the word S(xl)S(x2)... S(Xk) wraps above ~, and it is closed. 
Therefore 
0= H(S(x,)S(x2).. .S(Xk)) by (i) 
= n(S(xa)) + H(S(x2) )+ ' . .  + H(S(Xk)) 
= -A(p)x~ - A(ft)x2 . . . . .  A(~)Xk by (ii) 
= -A(p)X(P) ,  
which proves the first part of the theorem. 
If it* is a cyclic permutation, let us consider some v e V and, after an eventual 
rotation, let it* = V+Xk...X2X~v-M, with letters Xl, Xz,. • . ,  Xk e V + U V-  and a 
subword M. Following (iii) the word S(v-)S(xl)S(x2).. .  S(xk)S(v +) "wraps above 
/~. Therefore the first letter of S(xl) is equal to v +, and the last letter of S(Xk) is 
also equal to v ÷. Thus after removing either the first letter or the last letter of 
S(xl)S(xz).. .S(Xk) we get a closed word wrapping above ~. Property (i) 
implies v += H(S(xl)S(x2)...S(Xk)), which implies as above v+=-A( f t ) (x l  + 
x2+""  +Xk). But - (x~+x2+ " ' "  +Xk) is equal to the v-column of A(/z*). 
Therefore A(~)A(~*) is the identity matrix. [] 
Corollary 1 (Jaeger [5]). I f  G is an alternance graph whose adjacency matrix A 
considered over GF(2) has an inverse A -a, then A-1 is the adjacency matrix of an 
alternance graph G*. 
206 A. Bouchet 
Proof. Use the second part of the theorem with matrices considered modulo 2, 
G=G(m), G* =G(m*),  where m* is the double occurrence word whose 
separation is/z*. [] 
Corollary 2. The orientation of D(l~) is unimodular. 
Proof. Each matrix A[W] has an integral inverse when the inverse exists. [] 
There are two basic problems on unimodular orientations: (P1) to recognize 
whether a given orientation of a simple graph G is unJmodular; (P2) to recognize 
whether a simple graph G admits an unimodular orientation. Let us consider the 
case where G is bipartite with chromatic lasses V' and V". If A = (A~w: v, w e 
V(G)) is the adjacency matrix of G provided with an orientation, and 
B = (Bo.~.: v' • V', v" • V"), then it is easy to verify that the orientation of G is 
unimodular if and only if B is totally unimodular. Let M(G, V') be the binary 
matroid with a base equal to V' and the set of fundamental circuits {{v'} O 
{v": v'v" • E(G)}: v' • V'} with respect o V'. It is easy to verify that G can be 
provided with an unimodular orientation if and only if M(G, V') is a regular 
matroid. Thus Problems (P1) and (P2) are well solved (Camion [3], Tutte [9]) 
when G is bipartite. 
Problem (P2) is of special interest because Corollary 2 says that an alternance 
graph admits an unimodular orientation. It can be verified directly that every 
nonalternance graph of lowest order, 6, does not satisfy this necessary condition. 
To see that the condition is not sufficient we can use the following theorem of de 
Fraysseix [5]: a bipartite graph G with a chromatic lass V' is an alternance graph 
if and only if the matroid M(G, V') is graphic and cographic. Figs. 2 and 3 depict 
two bipartite graphs G2 and G3 where V' is made of the circled vertices. We 
Fig. 2. 
Unimodularity and circle graphs 207 
¥ * z 
Fig. 3. 
verify that M(G2, V') is the Fano matroid. This matroid is not regular, so that 
both de Fraysseix theorem and our necessary condition imply that (32 is not an 
alternance graph. M(G3, V') is the cycle matroid of K5 (V' corresponds in K5 to 
a tree made of the four edges incident o a same vertex). This matroid is regular 
but it is not a cographic matroid, so that Ga is not an alternance graph by de 
Fraysseix theorem when it admits an unirnodular orientation. 
The local complementation f a simple graph G at a vertex v is the operation 
which consists in replacing the subgraph induced on {w: vw e E(G)} by the 
complementary Subgraph. A graph is locally equivalent to G if it is obtained 
through successive local complementations starting with G. If v is a letter of the 
double occurrence word m, and we decompose m as AvBvC with suitable 
subwords A, B, C, and we replace the subword B by its mirror-image B', then 
the alternance graph of rn' =AvB'vC is the local complement of G(m) at v. 
Therefore any graph locally equivalent o an alternance graph is also an 
alternance graph. 
To see that (73 is not an alternance graph we can make local complementations 
at the vertices x, y, z, obtaining so a graph G~. If we delete x, y, z of G~, we get 
(72 which has no tmimodular orientation. If a graph has an unimodular 
orientation, this holds also for every induced subgraphs. Therefore G~ has no 
unimodular orientation, and it cannot be a circle graph. This is also the case for 
G3 which is locally equivalent to G~. 
Conjectnre. A graph G is an alternance graph if every graph locally equivalent to 
G has an unimodular orientation. 
208 A. Bouchet 
References 
[1] A. Bouchet, Caract~risation des symboles crois~s de genre nul, C. R. Acad. Sci. Paris 274 (1972) 
724-727. 
[2] A. Bouchet, Greedy algorithm and symmetric matroids, Math. Programming, to appear. 
[3] P. Camion, Characterization of totally unimodular matrices, Proc. Amer. Math. Soc. 16 (1965) 
1068-1073. 
[4] S. Even and A. Itai, Queues, stacks and graphs, in: Theory of Machines and Computations 
(Academic Press, New York, 1971) 71-86. 
[5] H. de Fraysseix, Local complementations and interlacement graphs, Discrete Math. 33 (1981) 
29-35. 
[6] M.C. Golumbic, Algorithmic Graph Theory and Perfect Graphs (Academic Press, New York, 
1980). 
[7] F. Jaeger, On some aglebraic properties of graphs, in: Progress in Graph Theory, J.A. Bondy and 
U.S.R. Murty eds. (Academic Press, New York, 1984). 
[8] W. Naji, Reconnaissance d s graphes de cordes, Discrete Math. 54 (1985) 329-337. 
[9] W. Tutte, Lectures in matroids, J. Res. Nat. Bur. Standards Sect. B 69 (1965) 1-47. 


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