On the X-join decomposition for undirected graphs

Discrete Applied Mathemati~ 1 (I979) 201--207 © North-Holland Publishing Company

ON THE X-JOIN DECOMPOSITION UNDIRECTED GRAPHS

FOR

M. H A B I B

Insfitut de Programmafion, 4, place Jussieu, 75230 Pa

~edex 05, France

and

M.C. MAURER CNAM-IIE, 292, rue Saint-Martin, 75141 Paris Cedex )3, France Received 6 Febi~miS, 1978 Revised 17 April 1979 In his paper [17], Sabidussi defined the X-join of a family of graph~. Cowan, .tames, Stanton gave in [6] an O(n *) algorithm that decomF.oses a graph, when possible, into )ht~ X-join of the family of its subgraphs. We gwe here another approach using an equivalence irelation or the edge set of the graph. We prove that if G ~,nd its complement are eonneete,.~th~enthere exists an unique class of edges that covers all the vertices of G, This theorem yields i nmediately an O(n 3) decomposition algorithm.

t. Introduction In t h e p r e s e n t paper, we deal with finite undirected, loopless gra11~hs w i t h o u t multiple edges. T h e v e r t e x set a n d t h e e d g e s e t of a g r a p h G will be d e n o t e d V(G) and E(G) respectively. F'~r u ~ V(G), we denote N(u)= {v~-V(G)]uv~E(G). a n d for A ~ V(G) we d e n o t e GA the subgxaph of G w h e r e V ( G A ) = A , a n d E ( G A ) is t h e s u b s e t of E ( G ) m a d e u p of e d g e s with b o t h e n d s in A . L e t P be any p a r t i t ' a n of V(G), t h e n ~[re q u o t k a t g r a p h of G m o d u l o P is t h e g r a p h G/P with V(G/P) = P a n d E(G/P) = { A B i A~ B ~ P, A # B s u c h that t h e r e exist a ~ A , b e B with a b l E ( G ) } . Definition 1, A s u b s e t A of V ( G ) is e~:ernally related in G, if

N(x)-A=N(y)-A

Vx, y ~ A .

Obviously: 0, {x} for every x ~ V(G), a n d V ( G ) are such subsets. Definition 2. L e t P c { V 1 . . . . . Vk} b e a partition of V(G). P is a pamtion of G if V i e [ l , k], Vi is externally related in G. 201

M. Habib.M.C. Maurer

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Clearly, if P '.'sa partition of G, then P is also a partition of (~, the complement of G. We denote 0G, 1~ respectively the following trivial partitions of G:{[x}, x e V(G)} an6 tV(Gy). We also use the following partial order between partitions: V---{e, . . . . . P~}~
p~cQi.

The aim of the presenlt work is the search of non-trivial maximal partitions, in some special sense, of a given graph G if one exists. We now mention the equivalence between the above notations, and sor~';e other notations of the same concept that could be found in literature. P = iV~ . . . . . Vk} is a partition of G: -ItI G is the G/P join of the family Gv, in the sense of Sabidussi [17], Sumner [19], Hemminger [12], Jolivet ~13]. -Iff the equivalence: binary relation on V(G) associated with P is a U(G)congruence with U(G)={(x,y) and (y,x) when xy~E(G)} as denoted by Chatelet [4]~ Pfaltz [16] and Eftimie [8]. -lff v~ - - ~,,r:/P~'2' ~,,...o,~ with the substitution notati. ~ avatal [5] and Balas, Zemel [3]. 2. l~actorizatJ,on theoremt~

Definition 3. A graph is igrime if the only partitions it admits are trivial. Defmilion 4. A graph G is frctgile if G or t~ is disconnected. Furthermore ff G (respectively G) has !,. :-1 connected components that induce a partition P= {V1 . . . . . Vk} of G, we denote G =Y..~=1Gv. (resp. G --=I'[~=1Gv,). And in bolth cases, though P is not the maximal non-trivial partition of G in the usual meaning, we shall call P the maximal partition of G. Remark 1. The following graph G = (~1, 2}, ~12}) and its complement are the only two graphs which are both fragile and prime. By convention, we consider them as prime. These definitions and conventions lead us to factorization theorems. Theorem 1. Every graph G has a unique maximal partition distinct from 1~. Such a p~r~ition will be called for short the maxhnal partition of G, and we can observe that, when G i;~ prime, this partition is 0G. This theorem 1 has appeared several times and in particuiar can be found in the work of Billera [2], Cunninlgham [7], Gallai [10], Maurer [14], Shapley [18]. Remark 2, If G is not a fragile graph, then P is its maximal partition iff G/P is prime. Besides, if P={VI . . . . , Vk} is the maximal partition of a given graph G, the subgraphs Gv. are not ~ecessarily prime, so we can test if G could be further decomposed using the most natural rec~rsive process.

On the X-loin decomposiaon

203

Let G be a graph: ff G is prime, then P = 0G, end. oiher~llse let P~ ={V~ . . . . . Vk} be the maximal partition of G, then f ' = {P(Gv,) . . . . . P(Gv~)I where P(Gv,) = lc~, if Gv, is prime and, othermlse P(Gv,) = {F(Gv,,) . . . . P(Gv~)} when {V~ . . . . . V~} i; the rr x i m a l partition of Gv,.

Theorem 2. This process yields a parti~on P = v . . . . . Vp} where all subgraphs Gv, are prime, and furthermore P is the unique m,~::ir, 'at partition ~'ith this property. This partition will be called the maximum ord,,~r artition of G.

We shall use now a binary relation F, pointed out by Golumbic [11]. F is defined on the edge set E ( G ) of a graph G as folllows: a b F a c if[ b---c or bc~ E(G). F', the transitive closure of F, partitions U(G) into classes that we shall call F-classes of E(G) (these F-classes have a great importance when one looks at the transitive orientations of comparability graphs). Definition 5. A F-class a of E ( G ) covers a vertex x if there exists y e V ( G ) - { x } , such that xy ~ a. Definition 6. The sequence of edges ab = a o b o F a l b l F ' " Fakbk = cd is called a F-chain from ab to cd. Let us recall two useful properties of F-classes.

Property 1. (Arditti [1]). The subset of V(G) covered by a F-class t~ is externally related in G. Furthermore, if G has at ~east two F.-classes, then one of these aoes not cover V(G) itself. Properly 2. (Ardhti [1]). Let Vo be an externally related set in G, such that there exists e ~ E(Gvo); then, r e ' E E(G), e'l 're implies e' ---xy with x e Vo, y ~ Vo. The above properties give us an algorithmic approach to prime graphs (in O(]V(G)la), see Section 4). To this end, we have to specify some particular externally related sabsets.

204

M. Habib, M.C. Maurer

DefmltioR 7. Let x, y c: V(G) such that {x, y) is externally related in G; x, y are externally (resp. totally) equivalent when xy~ E(G) (resp. xy ~ E(G)), Thqmrem 3. Let G be a graph witho~t isolated vertices such that I~'(G)I > 3, then the following are equivalent: (i) G is prime, ~ii) G has an u1~,ique F-class and no totally equivalent vertices, (iii) G and G have a uniql~e .E-class, Proo|. We observe that with such a graph G, if there exists Vo, a non-trivial externally related subset~ there, as G has an unique F-class, hence E(Gvo)= 0 using Property 1. Although we know very little about prime graphs, we have a very practic~fl nece~.sary conditiorl due to Foulis [9]. Prope~¢ 3, Let G be a prime graph with IV(G)[ > ? then ~ contains a subgraph isomorphic to the hook (the chain of length three). We can now demonstrate the main theorem. Theorem 4. If G is not a fragile graph, then there exist,: ~ unique F-class that covers V(G). l?roof. Let P be the maximal partition of G. Since G is not fragile one has IV(G/P)[ > 3 and therefore by Property 3~ G/P contains a hook H. Furthermore, it is very easy to see tha~ all the edges of G corresponding t3 those of H are in the same F-clas~ ~. But G/P is prime (by Remark 2) and so, has a unique F-class; thus, for every edge c ~ E(G/P) there exists a F-chain in G/P trom e to e', e' ~ E(H). Then, following this F-chain, all the corresponding edges of G are also in ct. We conclude that all edges of G induced by those of G/P are i.n the same F-class 0t, and the desired conclusion follows immediately. Corollary. Under the same hypothesis, the elements of ~he maximal par~,itio~.tof G, are the classes of totally equivalent vertices of the graph G' with V(G') = V(G) a~',d E(G') consisting of the edges bei'onging to the unique F-class that cotters '¢(G). The proof is obvious frem Theorem 4, but this corollas3, leads immediately to a procedure.

4. Decomposition algorithm l¢laximal partition procedure (a) When G is a fragile graph, the connected components of G or (3 give the maximal partition P, end.

On the X-ioin decomposition

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(b) Otherwise, determine the/'-classes of G. (c) It G has at least two F-classes, then construct the graph G'. (d) Determine the classes of totally equivalent vertices in G ' : they give the elements of P, end. Kemark 3. When the maximum order partition is required, we can apply this procedure recursively, observing that it is only necessary to determine the F-classes once (see Properties 1, 2). To compute the F-classes of a given gr~ ,1 G we can use the iollowing algorithm. F-classes algorithm Let G be a non-fragile graph, with : ( G ) = { ~ , x 2 . . . . . x,}, E ( G ) = {e~. . . . . e,,}. C is an array of ler~gth m, whose el, nents i~:dicate the F-cla:sses (e.g. C(e~) = a iff the edge e~ is in the ck,ss a ) tL is a .st containing all the edges of the F-class ~. lnilia~ly:Vi ~ [1, m]C(e~) ,,-- i; P~ *-- e~; For i varying fr3m 1 to n :comoute the connected components Gj, j ~ [ 1 , k~], V(G~) = {x~. . . . . x~,}, of the subgraph (~N¢~) of t~. For j varying from 1 to k~: h = Lj t~=l

VL ~ [2, Li], C(x[, xi) "" C(x~, xi) End.

Property 4. The p evious algorithm gives the F-clas~es of G. Proof. This algorithm is made up with n steps, each c,f them consist~ in merging the lists Pk locally around a vertex x~, -vith respect to t~e relation F. Let us denote this operation r/z, a:,d C~, Cr respectively, the initial ~ncl final partitions of E(G) of this algorithm, then Cf= ~ . , ( . " "Ox,(C~)" "). If F t partitions E(G) in Cr, then trivially Cr~ Cr. Furthermore, the operations 7/~ commute and are idempotent, so we have C~ = C~. Theorem 5. The number of operation:: required to comp~'~te the maximal partition or the maximum order partition is O(IV~ (~)[3). ProoL The steps (a), (c), require a namt~er of operations in O([E(G)I) and the step (d) in O([V(G)i2). So the F-classes algorithm dominates, as it requires O([V(G)I-') operations. This result is the same for the Maximum order partition, as we have only to repeat the steps (a), (c), (d), at most n times.

M. Habib,M.C. Maurer

206

In fact, thc notions of externally related subsets, partitions of a graph, prime graphs, are easily translatable for directed graphs and Theorems 1, 2 still work. Nevertheless, we have not yet found a binary relation that would play the role of F, consequently the only algorithm we have re-quffes O(]V(G)4]) operations; fi3r details see Maurer [15]. This algorithm could be used to minim~e storage in the sense of graph structure (see Pfaltz [16]). For some problems, path finding for example, we can represent a graph G b5 preserving only the graphs GIP and Gv,, i E.-"El, k], wi'th P = {V~. . . . . ',/~} the maximum order partition of G. Another application is for isomorphism problems:in fact, it is easy to verif~ ~hat if G~G2 are isomorphic, then G,/P,, C2/P2 are also isomorphic, when ~, esp. P2) is the maximal partition of G~ (resp. G2).

5. Automorphisms

We call automorphism of a graph G, a 1 - i mapp'ng from V(G) onto itself preserving adjacent.y, and let us denote by Aut (G) the automorphism group of G. (Trivially every V' c Aut (G) generates/2, a 1-1 mapping from E(G) to itself). Definition 8, If P = {V~ . . . . . natural for P if YiE[1. k],

Vs~} is a partition of G and if v~~ A u t ( G ) /x is

3 j ~ [ 1 , k]

~(Gv,)=Gv,

or in other words ~ ( P ) = P. Such an automorphism induces g * e Aut (G/P). Property/5. Let/~ ~ Aut (G) and a be a F-class then/2(a) is also a F-.clar~ of G. Proof. The relation F is preserved by every # ~ Aut (G), and thus the F-classes

also. Theorem 6. Let P={V1 . . . . . V~} be the maximal partigon of G. then every

e Aut (G) is natural for P. Proof, Let us suppose first that G is not a fragile graph, then ~:here exists a unique F-class a that covers V(G); necessarily, we have 12(a) = a. And w'~th the corollary of Theorem 4, we easily obtain tz(P)= P. In the other case, when G is not connected for example, as mentioned in Definition 4, P, the maximal partition of G is the partition induced by its co mected components; as p. preserves edgeconnectivity, Iz(P)= P is still valid. Furthermor~., there is no difficulty to show that every t~ ~ Aut (G) is natural for every partition Q of G obtained in the following way. There exists a sequence of

On the X-~oin de.composition p a r t i t i o n s of G , f r o m F t h e m a x i m a l p a r t i t i o n of G , t o O : po = p, p s . . . . . s u c h t h a t if P~ = { V ~ . . . . .

V~}, t h e n *r , ~

207 p,, = Q,

= t "v" . ) , / h ~ ~ ] } U P t for s o t n e j e [ 1 , ~ l ,

and

P~ is t h e m a x i m a l p a r t i t i o n of Gv,, U n f o r t u n a t e l y , w e h a v e n o t y e t gc~t a r, re c h a r ~ . c t e r i z a t i o n of t h e s e t of p a r t i t i o n s of G , f o r w h i c h e v e r y ~ e A u t ( G )

natural,

Ret~rences [1] J.C. Arditti, Graphes de comparibilit(: et dimcnsio.-, :s ordres, Note de recherches CI~',M 607. Centre de Recherches Math~matiql~es de l'Unlversi~ de Monlrdal ',1976). [2] L.J. BiUera, On the composition and decomposition of clutters, J. Combinatorial Theory (B) 11 (lO71"l 234-245. [3] E. Balas, E Zemel, Graph substitution and set packing polytope~, Networks 7 (1977) 267-2h ~. [4] A. Chateiet, Alg~bre des relations de congnlence, Ann. Scietltifiques de l'Ecole Normale Sup~rieur; 66 (1947) 337-368. [5] V. Chvfital, On certain pol~opes associated with graphs, J. Combinatorial Theory (B) 18 (1975) 138-15'~. [6] D.D. Cowan, L.O. James and R.G. Staoton, Graph d~composition for undirected gra~ :s, in: F. Hottrnan, R.B. Levow, eds., 3rd South-Eastern Conf. Combinatorics, Graph Them'y, and Computip, g (Utilitas Mathematica, Winnipeg, 1972) 281-290. [7] W.H. Cumlingham, A combinatorial decomposition theo~, PhD. Thesis, University of Waterioo, Ontario ,'J 973). [8] M. Eftimie a;~d R. Eftimie, A decomposition properly of basic acyclic graphs, D~screte M~thematics 17 (1977) 27t-279. [9] D.J. Fou!is, Emv rical Logic, Xeroxt~d course notes, University of Massachusetts, Amherst. Mass. (1969/70). [10] T. Galla~, Traeslt~, orienterbare Gra~hen, Acta Math. Acad. Sc;.. Hung., 18 (1967) 25-66. [11] M.C. G,.~lnmbic, Comparability graphs and a new matroid, 2. Combinatorial Theory (B) 22 (19;7) 68-90. [12] R.L Hemminge, The group of an X-join of graphs, J. Combin~.iorial Theory 5 (196,',) 408-418. [13] J.L. Jolivet, ProLl~mes de connexit~ et bamiltonienc en th~orie des gra,~hes et gdn6ralisztion de la motion de graphe parfait, T~'~se Sci. Math. Universit~ Paris VI (1975~. [14] M.C. Maurer, U:~it6 de la ddcomposition d'un graphe en joint suivant un graph¢ jointirr~duetible, d'm e famine de ses sour-rraphes, C.R. Aead. Sci. Paris t. 282, s~rie A (197b) 2~q9-202. [15] M.C. Maurer, Josnts et ddcompositions ~'remibres dans les graphes, th~sc 3~me cycle, Universit~ Paris VI (1977). [16] J.L PfalLz, Graph structures, J. Assoc. '?ompul. Mach. 19 (1972) 411-422 [17] G. Sabidussi, Graph derivatives, Math :'eitschr. 76 (1961) 385-401. [18] L.S. Shapley, Ccmpou-d smrple games ~U:on committees, the RAND Corpolation, PAVl-5438PR (October 1967). [19] D.P. Sumner, Grapil undecomposable with respect to the X-ioin. Discrete Mathematics 6 (1973) 281-298.