The graph sandwich problem for 1-join composition is NP-complete

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1 Discrete Applied Mathematics 121 (2002) The graph sandwich problem for 1-join composition is NP-complete Celina M.H. de Figueiredo a;, Sulamita Klein a, Kristina Vuskovic b a IM and COPPE, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, Rio de Janeiro, RJ, Brazil b School of Computing, University of Leeds, Leeds LS2 9JT, UK Received 17 May 1999; received in revised form 13 April 2001; accepted 26 April 2001 Abstract A graph is a 1-join composition if its vertex set can be partitioned into four nonempty sets A L; A R; S L and S R such that: every vertex of A L is adjacent to every vertex of A R; no vertex of S L is adjacent to vertex of A R S R; no vertex of S R is adjacent to a vertex of A L S L. The graph sandwich problem for 1-join composition is dened as follows: Given a vertex set V, a forced edge set E 1, and a forbidden edge set E 3, is there a graph G =(V; E) such that E 1 E and E E 3 =, which is a 1-join composition graph? We prove that the graph sandwich problem for 1-join composition is NP-complete. This result stands in contrast to the case where S L = (S R = ), namely, the graph sandwich problem for homogeneous set, which has a polynomial-time solution.? 2002 Elsevier Science B.V. All rights reserved. Keywords: Computational complexity; Graph algorithms; Sandwich problems; Perfect graphs 1. Introduction We say that a graph G 1 =(V; E 1 )isaspanning subgraph of G 2 =(V; E 2 )ife 1 E 2 ; we say that a graph G =(V; E) isasandwich graph for the pair G 1 ;G 2 if E 1 E E 2. For notational simplicity in the sequel, we let E 3 be the set of all edges in the complete graph with vertex set V which are not in E 2. Thus every sandwich graph for the pair G 1 ;G 2 satises E 1 E and E E 3 =. We call E 1 the forced edge set, and E 3 the forbidden edge set. The GRAPH SANDWICH PROBLEM FOR PROPERTY is dened as follows [11]: GRAPH SANDWICH PROBLEM FOR PROPERTY Instance: Vertex set V, forced edge set E 1, forbidden edge set E 3. Corresponding author. addresses: celina@cos.ufrj.br (C.M.H. de Figueiredo), sula@cos.ufrj.br (S. Klein), vuskovi@comp.leeds.ac.uk (K. Vuskovic) X/02/$ - see front matter? 2002 Elsevier Science B.V. All rights reserved. PII: S X(01)

2 74 C.M.H. de Figueiredo et al. / Discrete Applied Mathematics 121 (2002) Question: Is there a graph G =(V; E) such that E 1 E and E E 3 = that satises property? Graph sandwich problems have attracted much attention lately arising from many applications and as a natural generalization of recognition problems [4,9 15]. The recognition problem for a class of graphs C is equivalent to the graph sandwich problem in which the forced edge set E 1 = E, the forbidden edge set E 3 = ; G=(V; E) isthe graph we want to recognize, and property is to belong to class C. Perfect graphs were introduced by Berge [1] as those graphs for which, in every induced subgraph, the size of a largest clique is equal to the chromatic number. The recognition problem for the class of perfect graphs is a famous open problem in computational complexity [16]. Golumbic et al. [11] have considered sandwich problems with respect to several subclasses of perfect graphs, and proved that the GRAPH SANDWICH PROBLEM FOR SPLIT GRAPHS remains in P. On the other hand, they proved that the GRAPH SANDWICH PROBLEM FOR PERMUTATION GRAPHS turns out to be NP-complete. We are interested in graph sandwich problems for properties related to compositions arising in perfect graph theory. Several compositions of graphs are known to preserve perfection: graph substitution [17], join composition [3,6,8], clique identication [2]. A graph G =(V; E) has a homogeneous set H; H V, if each vertex of V \ H is either adjacent to all vertices of H or to none of the vertices of H; H 2 and V \ H 1. Polynomial algorithms for nding homogeneous sets are given in [5,18 21]. A graph G =(V; E) isa1-join partition (or join partition) if its vertex set V can be partitioned into sets V L and V R so that V L 2 and V R 2, V L contains a nonempty set A L, V R contains a nonempty set A R, with the property that every node of A L is adjacent to every node of A R, no node of V L \A L is adjacent to a node of V R, and no node of V R \ A R is adjacent to a node of V L. Let S L = V L \ A L and S R = V R \ A R.If S L =, then A L is a homogeneous set in G, and if S R =, then A R is a homogeneous set in G. In [4], a polynomial-time algorithm is given for solving the graph sandwich problem when property is to contain a homogeneous set. To distinguish the property of containing a homogeneous set from the property of being a 1-join partition graph, we further impose the condition that S L and S R. In this case, we say that V L V R isa1-join composition of G, with left side V L, with right side V R, and with special sets S L ;A L ;S R ;A R. A polynomial-time algorithm, of complexity O(n 3 ), for 1-join composition recognition is given in [7]. In this paper, we prove that the graph sandwich problem when property is to be a 1-join composition is NP-complete, even when restricted to instances that admit a solution for the homogeneous set sandwich problem. 2. Proof of NP-completeness In this section, we prove that the GRAPH SANDWICH PROBLEM FOR 1-JOIN COMPOSITION is NP-complete by reducing the NP-complete problem 3-SATISFIABILITY to GRAPH SAND-

3 C.M.H. de Figueiredo et al. / Discrete Applied Mathematics 121 (2002) WICH PROBLEM FOR 1-JOIN COMPOSITION. These two decision problems are dened as follows: 3-SATISFIABILITY (3SAT) Instance: Set X = {x 1 ;:::;x n } of variables, collection C = {c 1 ;:::;c m } of clauses over X such that each clause c C has c = 3 literals. Question: Is there a truth assignment for X such that each clause in C has at least one true literal? GRAPH SANDWICH PROBLEM FOR 1-JOIN COMPOSITION Instance: Vertex set V, forced edge set E 1, forbidden edge set E 3. Question: Is there a graph G(V; E), such that E 1 E and E E 3 = which is a 1-join composition? Theorem 1. The GRAPH SANDWICH PROBLEM FOR 1-JOIN COMPOSITION is NP-complete. Proof. In order to reduce 3SAT to GRAPH SANDWICH PROBLEM FOR 1-JOIN COMPOSITION, we need to construct a particular instance (V; E 1 ;E 3 )ofgraph SANDWICH PROBLEM FOR 1-JOIN COMPOSITION from a generic instance (X; C) of3sat, such that C is satisable if and only if (V; E 1 ;E 3 ) admits a sandwich graph G =(V; E) which is a 1-join composition. First, we describe the construction of a particular instance (V; E 1 ;E 3 )of GRAPH SAND- WICH PROBLEM FOR 1-JOIN COMPOSITION; second, we prove in Lemma 2 that every 1-join composition G =(V; E) satisfying E 1 E and E E 3 = denes a truth assignment for (X; C); third, we prove in Lemma 3 that every truth assignment for (X; C) denes a 1-join composition G =(V; E) satisfying E 1 E and E E 3 =. These steps are explained in detail below.. Construction of particular instance of GRAPH SANDWICH PROBLEM FOR 1-JOIN COMPOSI- TION. The vertex set V contains: an auxiliary set of vertices {v L ;v;a L ;a R ;s L ;s R }; for each SAT variable x k ; 1 6 k 6 n, one corresponding vertex x k ; for each SAT clause c i ; 1 6 i 6 m, three vertices z1 i ;zi 2 and zi 3, and three sets of vertices V 1 i;vi 2 and V 3 i. As the notation of the auxiliary vertices suggests, the role of those vertices is to dene the sides of the remaining vertices in a 1-join composition of the graph we are about to construct. As we shall see, in any 1-join composition for this graph, vertices a L and s L have to be placed on the same side as vertex v L, vertices a R and s R have to be opposite to vertex v L. This property of the auxiliary vertices with respect to any 1-join composition will dene a forcing for the sides (left or right) of the remaining vertices of the graph. See Figs. 1 4, where solid edges denote forced E 1 -edges and dashed edges denote forbidden E 3 -edges. The forced edge set E 1 is the union of sets of edges: {a L a R ;s L a L ;a R s R }; {v L w: w m i=1 {zi 3 } {x 1;:::;x n ;s L ;a R ;v}}, {vw: w m i=1 {zi 1 ;zi 2 ;zi 3 } {x 1;:::;x n ;a L ;a R ;v L }}, m i=1 {x ia L ;x i a R ;z i 2 zi 3 ;a Rz i 1 ;a Rz i 2 ;a Rz i 3 ;a Lz i 3 }, m i=1 (Bi 1 Bi 2 Bi 3 ), where Bi 1 Bi 2 Bi 3 consists of auxiliary forced edges corresponding to each clause.

4 76 C.M.H. de Figueiredo et al. / Discrete Applied Mathematics 121 (2002) Fig. 1. Auxiliary graph for special instance (V; E 1 ;E 3 ). Fig. 2. Same side (a) and dierent side (b) gadgets respectively, for l i 1 = x k and for l i 1 =x k. Fig. 3. Dierent side (a) and same side (b) gadgets respectively for l i 2 = x k and for l i 2 =x k.

5 C.M.H. de Figueiredo et al. / Discrete Applied Mathematics 121 (2002) Fig. 4. Same side (a) and dierent side (b) gadgets respectively, for l i 3 = x k and for l i 3 =x k. The forbidden edge set E 3 is the union of sets of edges: {v L s R ;vs L ;vs R ;a R s L ;a L s R ; a L v L }, m i=1 {zi 1 zi 2 ;a Lz1 i }, m i=1 (Di 1 Di 2 Di 3 ), where Di 1 Di 2 Di 3 consists of auxiliary forbidden edges corresponding to each clause. In the sequel, for j =1; 2; 3; the auxiliary forced edges Bj i and the auxiliary forbidden edges Dj i will be detailed by considering six dierent cases. Let c i =(l i 1 li 2 li 3 ) be a SAT clause. We have two kinds of gadgets which we call, respectively, same side gadget and dierent side gadget. The name and role of these gadgets will become clear when we prove Claims 2 and 3 for Lemma 2 below. If l i 1 = x k, then the following nodes and edges are added to construct a same side gadget: V1 i = {ai 1 }, Bi 1 = {ai 1 a L;a i 1 a R;z1 i ai 1 ;v La i 1 ;vai 1 }, Di 1 = {zi 1 x k;a i 1 x k}. We say that x k and z1 i are connected by a same side gadget. On the other hand, if l i 1 =x k, where x k is the negation of variable x k, then the following nodes and edges are added to construct a dierent side gadget: V1 i = {ai 1 ;bi 1 ;qi L1 ;qi R1 }, B1 i = {s LqL1 i ;s RqR1 i ;a La i 1 ;a Ra i 1 ;a Lb i 1 ;a Rb i 1 ;ai 1 qi L1 ;ai 1 qi R1 ;bi 1 zi 1 ;v La i 1 ;v Lb i 1 ;v LqL1 i ;vai 1 ;vbi 1 ; vqr1 i }, Di 1 = {qi L1 x k;qr1 i x k;z1 i ai 1 ;ai 1 bi 1 }. We say that x k and z1 i are connected by a dierent side gadget. If l i 2 = x k, then the following nodes and edges are added to construct a dierent side gadget: V2 i = {ai 2 ;bi 2 ;qi L2 ;qi R2 }, Bi 2 = {s LqL2 i ;s RqR2 i ;a La i 2 ;a Ra i 2 ;a Lb i 2 ;a Rb i 2 ;ai 2 qi L2 ;ai 2 qi R2 ; b i 2 zi 2 ;v La i 2 ;v Lb i 2 ;v LqL2 i ;vai 2 ;vbi 2 ;vqi R2 }, Di 2 = {qi L2 x k;qr2 i x k;z2 i ai 2 ;ai 2 bi 2 }. We say that x k and z2 i are connected by a dierent side gadget. On the other hand, if l i 2 =x k, then the following nodes and edges are added to construct a same side gadget: V2 i = {ai 2 }, Bi 2 = {a La i 2 ;a Ra i 2 ;zi 2 ai 2 ;v La i 2 ;vai 2 }, Di 2 = {zi 2 x k;x k a i 2 }. We say that x k and z2 i are connected by a same side gadget. Finally, if l i 3 = x k, then the following nodes and edges are added to construct a same side gadget: V3 i =, Bi 3 =, Di 3 = {zi 3 x k}. We say that x k and z3 i are connected by a same side gadget. On the other hand, if l i 3 =x k, then the following nodes and edges are added to construct a dierent side gadget: V3 i = {qi L3 ;qi R3 }, Bi 3 = {s LqL3 i ;s RqR3 i ; z3 i qi L3 ;zi 3 qi R3 ;v LqL3 i ;vqi R3 }, Di 3 = {x kql3 i ;x kqr3 i }. We say that x k and z3 i are connected by a dierent side gadget.

6 78 C.M.H. de Figueiredo et al. / Discrete Applied Mathematics 121 (2002) Lemmas 2 and 3 prove the required equivalence for establishing Theorem 1. Lemma 2. If the particular instance (V; E 1 ;E 3 ) of GRAPH SANDWICH PROBLEM FOR 1-JOIN COMPOSITION constructed above admits a 1-join composition G =(V; E) such that E 1 E and E E 3 = ; then there exists a truth assignment that satises (X; C). Proof. Suppose there exists a sandwich graph G =(V; E) that has a 1-join composition V L V R, where V L denotes the side of the partition that contains the node v L. Let S L ;A L ;S R and A R be the special sets of this 1-join composition. Claim 1. s L S L ;a L A L ;s R S R ; and a R A R. Proof. We rst show that s L ;a L V L and s R ;a R V R by considering two cases. Case 1: v V L. Since {w: wv L E 1 } {w: wv E 1 } = V \{s R } and S R, we have S R = {s R }. Since s R S R and s R a R E 1, we have a R V R. Since {w: wa R E 1 } {w: ws L E 1 } {w: ws R E 1 } = V \{s L } and S L, we have s L V R and hence s L V L. Since a R v; a R v L E 1, we have v L ;v A L. Since a L v E 1 and a L v L E 3,we have a L V R and hence a L V L. Case 2: v V R. Since v L v E 1, we have v L A L and v A R. First suppose that s L V R. Since {w: wv E 1 } {w: ws L E 1 } = V \{v; s L ;s R } and S L, we have S L = {s R }. Since s R a R E 1, we have a R V L. Since a R v E 1, we have a R A L. But then s L v L E 1 and s L a R E 3 contradict the assumption that V L V R is a 1-join composition. Therefore, s L V L. Since s L v E 3, we have s L S L. Since a L s L E 1, we have a L V L. Suppose that a R V L. Since a R v E 1, we have a R A L. Since s R a R E 1 and s R v L E 3, we have s R V R and hence s R V L. But then, {w: wa R E 1 } {w: ws L E 1 } {w: ws R E 1 } = V \{s L } implies that S R =, which contradicts the assumption that V L V R is a 1-join composition. Therefore a R V R. Since a R v L E 1, we have a R A R. Since s R a R E 1 and s R v E 3, we have s R V L and hence s R V R. Therefore, a L ;s L V L and a R ;s R V R. Since a L a R E 1, we have a L A L and a R A R. Since s L a R E 3, we have s L S L. Since s R a L E 3, we have s R S R. This ends the proof of Claim 1. Claim 2. For i =1;:::;m and j =1; 2; 3; if zj i and x k ; for some k {1;:::;n}; are connected by a same side gadget; then zj i and x k are on the same side of 1-join composition V L V R of G. Proof. By Claim 1, a L A L and a R A R. Since x k a L ;x k a R E 1, we have x k A L A R. First suppose that j = 3. Since z3 i a L;z3 i a R E 1, we have z3 i A L A R. Since z3 i x k E 3, we have that z3 i and x k must be on the same side of 1-join composition V L V R. Now assume that j = 1 or 2. Since a i j a L;a i j a R E 1, we have a i j A L A R. Since a i j x k E 3, we have that a i j and x k must be on the same side of 1-join composition V L V R. Since zj iai j E1 and zj ix k E 3, we conclude that zj i must be on the same side of 1-join composition V L V R as a i j and x k. This ends the proof of Claim 2.

7 C.M.H. de Figueiredo et al. / Discrete Applied Mathematics 121 (2002) Claim 3. For i =1;:::;m and j =1; 2; 3; if zj i and x k ; for some k {1;:::;n}; are connected by a dierent side gadget; then zj i and x k are on dierent sides of 1-join composition V L V R of G. Proof. By Claim 1, s L S L, a L A L, s R S R and a R A R. Since s L S L and qlj i s L E 1, we have qlj i V L. Since s R S R and qrj i s R E 1, we have qrj i V R. Since x k a L ;x k a R E 1, we have x k A L A R. First suppose that j = 3. Since z3 i a L;z3 i a R E 1, we have z3 i A L A R. Since qr3 i zi 3 E1 and qr3 i x k E 3, it follows that z3 i and x k cannot both be in V L. Since ql3 i zi 3 E1 and ql3 i x k E 3, it follows that z3 i and x k cannot both be in V R. Hence, z3 i and x k are on dierent sides of 1-join composition V L V R. Now assume that j = 1 or 2. By a similar argument as above, x k and a i j must be on dierent sides of 1-join composition V L V R. By a similar argument as in Claim 2 applied to vertices a i j ;bi j and zi j, we have that ai j and zi j must be on the same side of 1-join composition V L V R. Hence, x k and zj i must be on dierent sides of 1-join composition V L V R. This ends the proof of Claim 3. We now dene the following truth assignment for (X; C): for k =1;:::;n; x k is false if and only if vertex x k V L. By the construction of (V; E 1 ;E 3 ) and by Claims 2 and 3, for i =1;:::;m, literal l i 1 is false if and only if zi 1 V L, literal l i 2 is false if and only if z2 i V R, and literal l i 3 is false if and only if zi 3 V L. Suppose that for some i {1;:::;m}, the clause c i =(l i 1 li 2 li 3 ) is false. Then zi 1 ;zi 3 V L and z2 i V R.By Claim 1, a L A L and a R A R. Since z1 i a R;z3 i a R E 1, we have z1 i ;zi 3 A L. But then, since z1 i zi 2 E3 and z2 i zi 3 E1, the assumption that V L V R is a 1-join composition of G is contradicted. Hence, the above dened truth assignment satises (X; C). This ends the proof of Lemma 2. The converse of Lemma 2 is given by Lemma 3. Lemma 3. If there exists a truth assignment that satises (X; C), then the particular instance (V; E 1 ;E 3 ) of GRAPH SANDWICH PROBLEM FOR 1-JOIN COMPOSITION constructed above admits a 1-join composition G =(V; E) such that E 1 E and E E 3 =. Proof. Suppose there is a truth assignment that satises (X; C). We shall dene a partition of V into sets V L ;V R that in turn denes a solution G for the particular instance (V; E 1 ;E 3 )of GRAPH SANDWICH PROBLEM FOR 1-JOIN COMPOSITION associated with the 3SAT instance (X; C). Place vertices v L ;a L and s L in V L, and vertices v; a R and s R in V R. For k =1;:::;n, if x k is false, then place it in V L, and otherwise in V R. For each clause c i ;i=1;:::;m, place the remaining vertices as follows: If l i 1 = x k, then place z1 i and ai 1 on the same side as x k.ifl i 1 =x k, then place ql1 i in V L;qR1 i in V R, and a i 1 ;bi 1 and zi 1 on a dierent side from x k. If l i 2 = x k, then place ql2 i in V L, qr2 i in V R, and a i 2 ;bi 2 and zi 2 on a dierent side from x k. If l i 2 =x k, then place z2 i and ai 2 on the same side as x k. If l i 3 = x k, then place z3 i on the same side as x k.ifl i 3 =x k, then place ql3 i in V L;qR3 i in

8 80 C.M.H. de Figueiredo et al. / Discrete Applied Mathematics 121 (2002) V R and z3 i on a dierent side from x k. Note that the above placement of vertices z1 i ;zi 2 and z3 i ;i=1;:::;m, implies that for each clause c i;i=1;:::;m, literal l i 1 is false if and only if z1 i V L, literal l i 2 is false if and only if zi 2 V R, and literal l i 3 is false if and only if z3 i V L. Sets S L and S R are dened as follows: Place s L in S L and s R in S R. For i =1;:::;m and j =1; 2; 3: if zj i is connected by a dierent side gadget, then if zi j is in V L place qlj i in S L, and if zj i is in V R place qrj i in S R. For i =1;:::;m,ifz1 i is in V R, then place z1 i in S R, and if z2 i and zi 3 are both in V R, then place z2 i in S R. Let A L = V L \ S L and A R = V R \ S R. Let E = E 1 {uy: u A L and y A R }. To show that the graph G =(V; E) is a sandwich graph with 1-join composition V L V R, we need to show that the following types of edges do not exist: uy E 3 such that u A L and y A R, uy E 1 such that u S L and y V R, uy E 1 such that u V L and y S R. Let e E 3 and suppose that one endnode of e is in A L and the other is in A R. Then neither s L nor s R is an endnode of e. Edge e cannot be of the form a L z1 i, for some i {1;:::;m}, because a L A L and if z1 i is in V R then z1 i is in S R. Edge e cannot be of the form z1 i zi 2, for some i {1;:::;m}, because that would imply that zi 1 is in V L (since if z1 i is in V R then it is in S R ) and so z2 i is in A R, which means that z3 i is in V L (since z2 i and zi 3 are both in V R then z2 i is in S R), which would imply that all three literals in c i are false. Let i {1;:::;m}. Ifz1 i and x k are connected by a same side gadget, then e D1 i since x k;z1 i and ai 1 are all placed on the same side. If zi 1 and x k are connected by a dierent side gadget, then a i 1 ;bi 1 and zi 1 are all placed on the same side (hence e a i 1 bi 1 ;zi 1 ai 1 ), zi 1 and x k are placed on dierent sides, so if x k V L then qr1 i S R (hence e x k ql1 i ;x kqr1 i ), and if x k V R then ql1 i S L (hence e x k ql1 i ;x kqr1 i ). Therefore e D1 i. Similarly e Di 2 Di 3. Let e E 1 and suppose one endnode of e is in S L and the other is in V R. If s L u E 1, then u = a L ;v L or qlj i for some i {1;:::;m} and j {1; 2; 3}. Since all these nodes are in V L, edge e cannot have s L as an endnode. Thus an endnode of e is qlj i for some i {1;:::;m} and j {1; 2; 3}. But then qi Lj S L, hence zj i V L and so e ql3 i zi 3 ;qi L2 ai 2 ;qi L1 ai 1. Let e E 1 and suppose that one endnode of e is in S R and the other is in V L. If s R u E 1, then u = a R or qrj i for some i {1;:::;m} and j {1; 2; 3}. Since all these nodes are in V R, edge e cannot have s R as an endnode. Suppose that for some i {1;:::;m} and j {1; 2; 3}, qrj i is an endnode of e. Then qi Rj S R, hence zj i V R and so e qr1 i ai 1 ;qi R2 ai 2 ;qi R3 zi 3. Suppose that for some i {1;:::;m}, zi 1 S R and that z1 i is an endnode of e. Then e zi 1 a R;z1 i v, since a R;v V R.Ifz1 i and x k are connected by a same side gadget, then e z1 i ai 1, since zi 1 and ai 1 are placed on the same side. If z1 i and x k are connected by a dierent side gadget, then e z1 i bi 1, since zi 1 and bi 1 are placed on the same side. Thus we may assume that for some i {1;:::;m}; z2 i S R and that z2 i is an endnode of e. Then zi 3 V R and hence e z2 i zi 3. Since a R;v V R,we have e z2 i a R;z2 i v.ifzi 2 and x k are connected by a same side gadget, then e z2 i ai 2, since z2 i and ai 2 are placed on the same side. If zi 2 and x k are connected by a dierent side gadget, then e z2 i bi 2, since zi 2 and bi 2 are placed on the same side.

9 C.M.H. de Figueiredo et al. / Discrete Applied Mathematics 121 (2002) Note that the particular instances (V; E 1 ;E 3 )of GRAPH SANDWICH PROBLEM FOR 1-JOIN COMPOSITION constructed in the proof of Theorem 1 always admit a solution for GRAPH SANDWICH PROBLEM FOR HOMOGENEOUS SET by dening H = {v L ;a L }. Therefore, we have established that GRAPH SANDWICH PROBLEM FOR 1-JOIN COMPOSITION is NP-complete, even when restricted to instances that admit a solution for the homogeneous set sandwich problem. Acknowledgements We are grateful to Luerbio Faria for helping us with the gures and to Marcia Rosana Cerioli for many conversations on sandwich problems. This research was done while the third author was visiting Universidade Federal do Rio de Janeiro supported by Conselho Nacional de Desenvolvimento Cientco e Tecnologico, CNPq. The rst two authors are partially supported by CNPq and PRONEX. We thank referees for their careful reading and valuable suggestions, which helped improve the presentation of this paper. References [1] C. Berge, Les problemes de coloration en theorie des graphes, Publ. Inst. Statist. Univ. Paris 9 (1960) [2] C. Berge, Graphs and Hypergraphs, North-Holland, Amsterdam, Translated from the French by Edward Minieka, North-Holland Mathematical Library, Vol. 6. [3] R.E. Bixby, A Composition for Perfect Graphs, Topics on perfect graphs, North-Holland, Amsterdam, 1984, pp [4] M. Cerioli, H. Everett, C.M.H. de Figueiredo, S. Klein, The homogeneous set sandwich problem, Inf. Process. Lett. 67 (1) (1998) [5] A. Cournier, Sur quelques Algorithmes de Decomposition de Graphes, These Universite Montpellier II, France, [6] W.H. Cunningham, A combinatorial decomposition theory, Thesis, University of Waterloo, Canada, [7] W.H. Cunningham, Decomposition of directed graphs, SIAM J. Algebraic Discrete Methods 3 (2) (1982) [8] W.H. Cunningham, J. Edmonds, A combinatorial decomposition theory, Canad. J. Math. 32 (3) (1980) [9] M.C. Golumbic, Matrix sandwich problems, Linear Algebra Appl. 277 (1 3) (1998) [10] M.C. Golumbic, H. Kaplan, R. Shamir, On the complexity of DNA physical mapping, Adv. in Appl. Math. 15 (3) (1994) [11] M.C. Golumbic, H. Kaplan, R. Shamir, Graph sandwich problems, J. Algorithms 19 (3) (1995) [12] M.C. Golumbic, R. Shamir, Complexity and algorithms for reasoning about time: a graph-theoretic approach, J. Assoc. Comput. Mach. 40 (5) (1993) [13] M.C. Golumbic, A. Wassermann, Complexity and algorithms for graph and hypergraph sandwich problems, Graphs Combin. 14 (3) (1998) [14] H. Kaplan, R. Shamir, Pathwidth, bandwidth, and completion problems to proper interval graphs with small cliques, SIAM J. Comput. 25 (3) (1996) [15] H. Kaplan, R. Shamir, Bounded degree interval sandwich problems, Algorithmica 24 (2) (1999) [16] D.S. Johnson, The NP-Completeness column: an ongoing guide, J. Algorithms 2 (4) (1981)

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