Markov bases for decomposable graphical models

Transcription

1 Bernoulli 9(6), 23, ISSN c 23 ISI/BS Markov bases for decomposable graphical models ADRIAN DOBRA Institute of Statistics Decision Sciences Department of Molecular Genetics & Microbiology, Duke University, Durham NC 28251, USA adobra@statdukeedu We show that primitive data swaps or moves are the only moves that have to be included in a Markov basis that links all the contingency tables having a set of fixed marginals when this set of marginals induces a decomposable independence graph We give formulae that fully identify such Markov bases show how to use these formulae to dynamically generate rom moves Keywords contingency tables; decomposable graphs; disclosure limitation; Gröbner bases; Markov bases; Markov chain Monte Carlo

2 Markov Bases for Decomposable Graphical Models Adrian Dobra Institute of Statistics Decision Sciences Department of Molecular Genetics & Microbiology Duke University Durham, NC 28251, USA June 8, 23 Abstract In this paper we show that primitive data swaps or moves are the only moves that have to be included in a Markov basis that links all the contingency tables having a set of fixed marginals when this set of marginals induce a decomposable independence graph We give formulas that fully identify such Markov bases show how to use these formulas to dynamically generate rom moves Keywords Contingency tables; Decomposable graphs; Disclosure limitation; Gröbner bases; Markov chain Monte Carlo; Markov bases 1 Introduction The problem of describing sets of multiway contingency tables that are induced by fixing a number of lower dimensional marginals has been the focus of many research efforts in recent years These sets 1

3 arise in a variety of contexts such as disclosure limitation (Dobra, 22; Fienberg et al, 1998; Fienberg et al, 22) the calibration of test statistics (Agresti, 1992; Diaconis Efron, 1985; Mehta, 1994) Diaconis Sturmfels (1998) proposed a general algorithm for generating rom draws from a set of tables with given fixed marginals Their approach is extremely appealing because, in theory, it can be used for arrays of any dimension Despite its generality, the power of this sampling procedure is limited because it requires access to a Markov basis a finite set of data swaps which allow any two tables with the same fixed marginals to be connected In addition to sampling, Markov bases can be employed to enumerate all the integer tables with a given set of marginals As a consequence, Markov bases allow one to create a replacement for a database consisting of a way contingency table, when such a replacement is needed to protect the individuals with rare characteristics whose identity might be disclosed by the release of a number of marginals (Willenborg de Waal, 21) Diaconis Sturmfels (1998) Dinwoodie (1998) suggest computing a Markov basis by finding a Gröbner basis (Cox et al, 1992) of a wellspecified polynomial ideal, but their method is difficult to employ even for tables with three dimensions because of the computational complexity of computing Gröbner bases The statistical theory on graphical models (Madigan York, 1995; Whittaker, 199; auritzen, 1996) shows that the conditional dependencies induced by a set of fixed marginals among the variables crossclassified in a table of counts can be visualized by means of an independence graph In particular, a lot of attention has been given to decomposable graphs (auritzen, 1996), a special class of graphs that can be broken into components such that (i) every component is associated with exactly one fixed marginal; (ii) no information is lost in the decomposition process, ie no marginal is split between two components Our aim is to show how graphical models could help us identify special settings in which we could 2

4 ( Z ( develop efficient techniques for considerably reducing possibly eliminating the amount of computations needed to identify a Markov basis After presenting some notation definitions in Section 2, in Section 3 we introduce decomposable sets of marginals discuss some of their properties In Section 4 we prove that primitive data swaps or moves are the only moves that have to be included in a Markov basis associated with a set of decomposable marginals give explicit formulas for dynamically generating such bases In the last section we make some concluding remarks 2 Data Swapping Markov Bases A table of counts is a dimensional array of nonnegative integer numbers Each variable,, recorded in such a table can take a finite number of values! "$% &$' A cell entry (*),, / et, in table is a nonnegative integer representing the number of units or individuals sharing the same attributes By considering an ordering of the cell indices in (eg, lexicographic), the multiway array can be transformed in a linear list of counts (ie, vector) et table with marginal cells 8 denote an arbitrary subset of 6 The marginal! of is the contingency 9 ;,! &;>= entries given by The marginals Two tables (?@),AB! EDF GIHAJ C (*),A KK are called overlapping if If all the counts in table TS U array?y NM PO, otherwise they are nonoverlapping are equal if all their cell entries are equal, in this case we write are zero, we write * with entries (?S),WX( ), *Z R with entries ($Y[),*\( ), RQ The sum of two tables ), Similarly, the difference between 3 ), 2 is another table is an

5 Z When moving table entries from one cell to the other, some of the cell entries could be increased other cell entries could be decreased, hence a data swap or move associated with is an array ), 3,DF containing integer entries, ie ), primitive move has two entries equal to, two entries equal to Z Z 3 Z, for all A, while the remaining entries are zero Intuitively, a move can be viewed as the difference between the postswapped the preswapped tables The table created by repeatedly applying data swaps to the original table is sometimes required to be consistent with the marginals that were previously made public (Willenborg de Waal, 21) Consequently, we are interested in data swaps that leave a number of marginals unchanged Definition 21 et,,, be subsets of 6 A move for that preserves the marginal tables specified by the index sets Denote by ),,,, is a data swap,,, ie Q, for all 3 the set of all tables with nonnegative elements that have their,, marginals equal to the corresponding marginals of A move is admissible for if belongs to ) 3 Since preserves the ) 3 if only if ) ),, for all Definition 22 A Markov basis is a finite collection of moves that preserve the connects any two way tables that have the same table that belongs to ),, there exists a sequence of moves,, marginals of, we have 3 marginals,, marginals In other words, for any,,, in such that, C T C T ) 3 (21) for "!$% "! Eq 21 says that the table is transformed into by employing moves in Since a Markov basis depends only on the index sets 4,, '&, we will say that is a Markov basis for

6 = ) 3, where ) 3 2 ) 3 is a table of counts Diaconis Sturmfels (1998) prove that a Markov basis for ) 3 always exists 3 Special Configurations of Marginals In this section we closely follow the notation definitions relating to graph theory introduced in auritzen (1996) Dobra Fienberg (2) A brief introduction with the basic graph terminology needed to underst the results that follow is given in Appendix A Consider a set of marginals,,, %& such that their index sets cover 6, ie 6 X We always assume that there are no redundant configurations in this sequence, that is there are no, such that We visualize the dependency patterns induced by by constructing an independence graph Each vertex in this graph represents a variable, We draw an edge between two vertices if only if the twodimensional array defined by the variables associated with these vertices is a marginal of some *,,, 6 Definition 31 The independence graph %) a graph with vertex set 6 P associated with edge set given by,,, %& is 2 ) for some oglinear models are the usual way of representing studying contingency tables with fixed marginals (Bishop 195) If the minimal sufficient statistics of a loglinear model define a decomposable independence graph, the model is said to be decomposable By analogy with loglinear models theory, we introduce decomposable sets of marginals 5

7 Definition 32 The set of marginals! independence graph %) sets associated with,k,,k &, ie,kk,,k & is called decomposable if its corresponding 3 is decomposable the cliques ) of are the index ) Therefore a decomposable set of marginals could represent the minimal sufficient statistics of a decomposable loglinear model,k,,k'& are consistent if, for any,, the ) marginal ofk%& is equal to the ) KM marginal of'& Definition 33 The marginals! M The consistency of a set of marginals does not necessarily imply the existence of a table having this particular set of marginal totals see, for example, lach (1986) To be more precise, ) could be empty even if,kk,,k & are consistent In the special case of consistent decomposable marginals, however, ) 3 is never empty (Dobra, 22) Decomposable sets of marginals have many other exceptional properties that have been well documented in the literature For example, the corresponding maximum likelihood estimates can be expressed in closed form (auritzen, 1996; Whittaker, 199) Additionally, Dobra Fienberg (2) obtain formulas for calculating sharp upper lower bounds for cell entries of tables in ) given that the marginals are consistent decomposable 4 Markov Bases for Decomposable Sets of Marginals The special structural properties of decomposable graphs can be further exploited to derive a Markov basis of primitive moves for ) ifk,k,,k%& is a decomposable set of marginals 6

8 The simplest decomposable graph has two vertices no edges This graph is the independence graph associated with the two oneway marginals of a twoway table It turns out that it is straightforward to describe a Markov basis in this case (Diaconis Gangolli, 1995; Diaconis Sturmfels, 1998) Proposition 41 Consider a twoway contingency table sums 3(, column sums ), Z 3( if ), if ), 3(*), chosen such that ), 9! "$ by, we define a table ), ), ), ), ), *W$ For some indices, with fixed row,,, ), otherwise (41), Then, 3 (42) is a Markov basis for the class of tables with fixed row sums Proof The oneway marginals of E are zero, hence, column sums will leave unchanged By making use of computational algebra techniques, Sturmfels (1995) gives a complete proof of the fact that the set of moves described in Eq 42 is indeed a Markov basis The number of moves in this Markov basis is The set of primitive moves we described above allows one to transform a given twoway table in any other twoway table with the same row column totals Proposition 41 is the starting point for developing Markov bases for an arbitrary decomposable graphical structure Consider the case of way contingency table with two fixed marginals! decomposable since it has exactly two cliques, K The corresponding independence graph %) is

9 We show that the Markov basis ) for ) is the union of the Markov bases of one or more twoway tables with fixed oneway marginals We distinguish two cases (A) If M with level sets This table has fixed row sums O, the two fixed marginals are nonoverlapping Introduce two new variables, respectively Take the twoway table will be the Markov basis of moves for the twoway table fixed column sums The basis) that crossclassifies as described in Proposition 41 for ) (B) Otherwise, if the two fixed marginals are overlapping, we have M O For every R J J J J J J K, define a table N( ),,D F G H J J with entries ( ), J J J J (*), This table has two fixed marginals T (? ),,DF J HAJ K J (? ), K,DF J HAJ J J J Case (A) shows how to construct a Markov basis for that J J J J preserves the two nonoverlapping marginals It follows that a Markov basis of moves for table that preserves the marginals Therefore ) ) J J D F J J is given by J J contains only primitive moves represents a Markov basis for ) (43) [Figure 1 about here] Example 1 Consider a fourway table with fixed threeway marginals The corresponding independence graph!!" In addition,! is represented in Fig 1 The edge!" S! are complete in, hence P S Y is a separator for is a decomposable graph with two cliques Consider the set of contingency tables I $( ), Y ), Y *W Y 9$ S 9 S 8

10 Y where ( basis ), Y (*), S Y For every table 3K), S ),, we know its row column sums S Y that leaves unchanged the oneway marginals of the table Y, respectively The Markov can be obtained as in Proposition 41 A Markov basis of primitive moves that preserves the marginals )A!!" ), S $ & S is the union We introduce the set of primitive moves associated with an arbitrary decomposable graph Definition 42 et ) let ) ) ) the set of cliques of a decomposable graph We be a tree having the Star Property on the set of cliques of For every edge, we consider the vertex sets as in Eq A1 The set of primitive moves associated with the decomposable graph is given by where ) By removing an edge ) think about ) ) was defined in Eq 43 3 D) from, we create two connected components ) as being the cliques of a graph with vertices ) /6 edges (44) We The tree result, & ) has the Star Property, hence U M or 5 separates from in As a is the independence graph of a decomposable model with two cliques we know that the set of primitive moves corresponding to is) Eq 44 essentially says that the set of primitive moves for a decomposable model with independence graph is just the union of the sets of primitive moves associated with the twoclique models induced by each minimal vertex separator of We have to show that Definition 42 is correct 9

11 Proposition 43 The set of primitive moves defined in Eq 44 is indeed a set of moves for the class of tables ) Proof et clique 3 ) ), we have either 3 Then or ) for some ) For any arbitrary Since ẄRQ K Q, it follows that we also have RQ Next we will state prove a series of results that will help us prove the main theorem of the paper Most of these propositions should be selfexplanatory However, it is worth mentioning the intuition that triggered them if we delete a vertex that belongs to exactly one clique from a decomposable graph, along with the edges incident to it, we obtain a graph that is still decomposable (Blair Barry, 1993) Consequently, by collapsing across a variable associated with such a vertex, all the conditional dependencies existent among the remaining variables are preserved The set of primitive moves associated with a twoclique model induces a set of primitive moves for a twoclique model embedded in it Collapsing across some of the variables not contained in both cliques preserves the structure of the moves in Eq 44 Proposition 44 et be a table with two fixed marginals * K The corresponding independence graph is decomposable has two cliques Consider a vertex set such that, The separator of is Define a map which assigns to every its ) marginal, ie )! Then the following are true (a) for any ), ) ) or )!PQ (b) the map ) ) is surjective (c) for every table ) every move )? ) 1 such that M ) (45)

12 there exists ) with )! ) (46) Proof To simplify the notation, assume that PO We consider the marginals with their associated vectors! across the variables in K indices et (a) In Proposition 41, we constructed ) column marginal A primitive move ),A ) We have,tk K The table can be obtained from,, along by collapsing by considering the twoway table with row marginal ) was obtained by choosing two row, two column indices Then the table is given by A! if ),A if ),A A A ), ), K otherwise ), ), T ), * ),A A * C DF J HAJ ) A We distinguish two cases (i) \ Since ), \ K! ),, we need to have T ), \ K! It follows that for Clearly, )! ), PQ if 3 Moreover, for, ), 3 Hence 11

13 = = = (ii) Thus )! index ), It follows that 3 ) ), if ),!X), otherwise (b) In order to prove that is surjective, we pick an arbitrary move It is easy to see that where (c) The move define the move ),8,DF J J ),! ), ), ) ) A! A )! is given by Z if ), if ),A ),A 3 AK A move in if ), ),8! ),A Z if K), choosing two indices For any? T, for Therefore we have to choose Then ), ), by if otherwise ) where \ otherwise ), ), ) We choose an such that is obtained by ), ), otherwise (4) in, the corresponding move defined in Eq 4 satisfies ), ), $ ), for every T), ),, such that ), $ ), ), Z ), $ ), ), Z 12 (48)

14 C In this case, Eq 46 holds From Eq 45, we obtain that ), ), ), ), (49) But ), ), ), C 8J H5JTDF J H5J K) Z Z (41) Eq 49 Eq 41 imply that hence there has to exist an index exist another index 9 J H5J DF J HAJ K) with ), K with ), T Similarly, there has to With this choice, Eq 48 holds Proposition 45 extends Proposition 44 to an arbitrary decomposable model Proposition 45 et be a table with fixed marginals * the set of cliques of a decomposable graph Consider a tree for Assume that the clique is terminal in let every ) ) ) ) ) 3 its marginal, ie! (a) for any, ) (b) the map is surjective on 3 (c) for every table there exists ) ),,K & such that )!2 X) T ) is having the Star Property Define a map which assigns to Then the following are true every move ) with )! ) 13 3 or )!/Q ) such that (411) (412)

15 ) ) Proof (a) Since the clique is terminal in, there exists a unique clique in ), say ) that The subgraph The set of primitive moves corresponding to the edge ) is ) By definition, RQ, hence )!PQ %) is decomposable ) ) ) X) from, with subtree obtained by removing from, ie et edge ) removing the edge ) assume that we always have ) By removing the same edge from the tree We define the vertex sets, ) ) X), we obtain the subtrees by, such that, assume et the Consider an arbitrary be the two subtrees obtained by Without restricting the generality, we ) W D D With this notation, according to emma A4, the tree %& K D will have the Star Property for the graph, consequently the set of primitive primitive associated with is )! ) D & Consider an arbitrary move ) 3 such that there must exist some edge ) ) such that, thus O In addition, we have 44, we obtain that ) ) ) 3 or )!/Q (b) In order to prove that is surjective on) ) From Eq 413, we see that there is an edge ) ) (413) From Eq 44, we see that ) We have By employing Proposition, we pick an arbitrary move in such ) 14

16 Z 4 that ) ) Since such that )!, Proposition 44 tells us that there must exist some (c) Again, Eq 413 shows that it is possible to find an edge ) ) This means that ) We have Proposition 44, we learn that there exists a move ) But we also have ) ) ) M ) M ) such that From ) 3 such that * ), hence Eq 412 is true We are now ready to present prove the main theorem of the paper Theorem 46 et be a decomposable graph with cliques ) % primitive moves ) * ) ) 3 Proof By induction If decomposes in ) with Z is a Markov basis for ) Then the set of 3 defined in Eq 44 is a Markov basis for the class of tables cliques, then we know from Proposition 41 that Suppose the theorem holds for any decomposable graph cliques We want to prove that the theorem is true for a decomposable graph with The original table is in the set ) 4 ) We have to show that there exist for terminal in et ) Denote its marginal, ie )! The marginals know that ) ) T 4 CT ) 3 Take an arbitrary table such that T, cliques (414) a tree having the Star Property for assume that the clique is Consider the map which assigns to every ) lie in the set ) is a Markov basis for ) 15 From the induction hypothesis we, so there exists a sequence of

17 Z 4 C 4 M 4 C 4 4 M moves for 3 4 sequence of moves ) Z C T such that 4 C T ) Proposition 45 tells us that the sequence of moves 3 4 in ) such that, for every 4 translates into another 4, we have 4, We obtain a table such that the marginals the marginals ) 4 CT ) 3, given by Z C T are the same Moreover, since we employed moves in ) %& %& are also equal, hence a series of moves 4 T 4 in ) for T 4 T ) which transform the table ) (415) (416) 3, This implies that we can find ) in ie From Eq 415, Eq 416 Eq 41 we obtain Eq 414, which completes the proof (41) Example 2 The graph in Fig 2 has 11 vertices 28 edges This is a decomposable graph with the set of cliques ) S, where Y!" 3,!" 3 S! 3 Y 2 " The tree on ) with edge set ) ) S ) Y & has the Star Property, therefore the separators of are *M X!" 3, S S!, Y P Y S " The set of primitive moves associated with is ) * ) S Y ) S 16 Y ) Y S,

18 Assume we are given an elevenway table with fixed marginals,tk,k K The independence graph associated with these marginals is Theorem 46 shows that ) is a Markov basis for ) S Y [Figure 2 about here] The family of Markov bases we identified is extremely appealing to the potential user since one doesn t even need to actually list the set of moves ) 3 Any Markov basis could grow extremely large due to the size of the original table, hence hling it might become quite problematic The procedure we outline below gets around this obstacle by dynamically generating moves in ) 3 The first step consists of computing the number of moves associated with every edge of the tree We uniformly generate a primitive move in ) by choosing an edge in with probability proportional to the number of primitive moves associated with it, then uniformly selecting a move from the set of primitive moves corresponding to the edge we picked ) ) ) ) ) Algorithm 4 et associated with be a tree having the Star Property for The set of separators will be given by ) "M for every ;4 et ) the edge ) ) do with ;4 / M from, let Consider the subtrees obtained by removing be the vertex sets associated with these subtrees, as defined in Eq A1 Calculate the weight 4 representing the number of primitive moves corresponding to the 1

19 edge ) B4 D = D = = end for Normalize the weights 3 To uniformly select a move in ), do 1 Romly select an edge ) with probability )$4E! B4, where ;4 / M 2 Uniformly pick a move in ), where were defined in Eq A1 5 Conclusions Many techniques that worked well for lowdimensional examples are almost impossible to use for problems that arise in practice due to the huge computational effort they usually require This paper demonstrates that graphical modeling is a very powerful tool for effectively overcoming major issues related to scaling up algorithms to make them suitable for use in highdimensional applications We represented the dependency patterns induced by a number of fixed marginals by means of graphs, by doing so, we identified Markov bases for an entire family of sets of tables We proved that a Markov basis for a decomposable model with cliques can be expressed as a union of Markov bases associated with ( Z ) models with two cliques Since the Markov basis of a model with two cliques is the set of primitive moves corresponding with one or more twoway tables with fixed oneway marginals, we deduce that the general decomposable case essentially reduces to the twoway case It seems important to point out that more results can be derived by exploiting techniques borrowed from the graphical models literature, namely decompositions of graphs by means of separators Dobra 18

20 Sullivant (23) have developed a divideconquer algorithm for significantly reducing the time needed to find a Markov basis when the underlying independence graph is not decomposable, but can be at least partially decomposed, though the resulting components of the decomposition may correspond to more than one fixed marginal Acknowledgments Preparation of this paper was supported in part by the National Science Foundation under Grant EIA to the National Institute of Statistical Sciences The author is extremely grateful to Dr Stephen E Fienberg Dr Stephen F Roehrig for all their support, comments valuable discussions The author would like to thank Seth Sullivant who provided comments on an earlier version of the paper References Agresti, A (1992) A Survey of Exact Inference for Contingency Tables (with discussion) Statistical Science,, Bishop, Y M M, Fienberg, S E, Holl, P W (195) Discrete Multivariate Analysis Theory Practice MIT Press, Cambridge, MA Blair, J R S Barry, P (1993) An Introduction to Chordal Graphs Clique Trees In Graph Theory Sparse Matrix Computation, ed IMA, ol 56, 1 3 Springererlag, New York Cox, D, ittle, J, O Shea, D (1992) Ideals, arieties Algorithms Springererlag, New York Dalenius, T Reiss, S P (1982) Dataswapping a Technique for Disclosure Control Journal of Statistical Planning Inference, 6,

21 Darroch, J N, auritzen, S, Speed, T P (198) Markov Fields oglinear Interaction Models for Contingency Tables The Annals of Statistics, 8, Diaconis, P Efron, B (1985) Testing for Independence in a TwoWay Table New Interpretations of the ChiSquare Statistic The Annals of Statistics, 13, Diaconis, P Gangolli, A (1995) Rectangular Arrays with Fixed Margins In Discrete Probability Algorithms, Springererlag, New York Diaconis, P Sturmfels, B (1998) Algebraic Algorithms for Sampling From Conditional Distributions The Annals of Statistics, 26, Dinwoodie, I H (1998) The DiaconisSturmfels Algorithm Rules of Succession Bernoulli, 4, Dobra, A (22) Statistical Tools for Disclosure imitation in Multiway Contingency Tables PhD thesis, Department of Statistics, Carnegie Mellon University Dobra, A Fienberg, S E (2) Bounds for Cell Entries in Contingency Tables Given Marginal Totals Decomposable Graphs Proceedings of the National Academy of Sciences, 9, Dobra, A Sullivant, S (23) A DivideConquer Algorithm for Generating Markov Bases of Multiway Tables Computational Statistics To appear Fienberg, S E, Makov, U E, Meyer, M M, Steele, R J (22) Computing the Exact Distribution for a Multiway Contingency Table Conditional on its Marginals Totals In Data Analysis from Statistical Foundations A Festschrift in Honor of the 5th Birthday of D A S Fraser, ed A, Saleh, Nova Science Publishers, Huntington, NY 2

22 Fienberg, S E, Makov, U E, Steele, R J (1998) Disclosure imitation Using Perturbation Related Methods for Categorical Data Journal of Official Statistics, 14, Haberman, S J (194) The Analysis of Frequency Data University of Chicago Press, Chicago auritzen, S (1996) Graphical Models Clarendon Press, Oxford Madigan, D York, J (1995) Bayesian Graphical Models for Discrete Data International Statistical Review, 63, Mehta, C (1994) The Exact Analysis of Contingency Tables in Medical Research Statistical Methods in Medical Research, 3, Sturmfels, B (1995) Gröbner Bases Convex Polytopes University ecture Series, American Mathematical Society ol 8 lach, M (1986) Conditions for the Existence of Solutions of the Threedimensional Planar Transportation Problem Discrete Applied Mathematics, 13, 61 8 Whittaker, J (199) Graphical Models in Applied Multivariate Statistics John Wiley & Sons New York Willenborg, de Waal, T (21) Elements of Statistical Disclosure Control ol 155, ecture Notes in Statistics Springererlag, New York A Graph Theory A graph is a pair ) 6, where 6 edges linking the vertices Our interest is in undirected graphs for which ) is a finite set of vertices 6 6 is a set of implies ) ; 21

23 For any vertex set 6, we define the edge set associated with it as et ) " ) if ) )W ) )W denote the subgraph of induced by Two vertices A set of vertices of 6 are adjacent is independent if no two of its elements are adjacent An induced subgraph %)W is complete if the vertices in are pairwise adjacent in We also say that is complete in A complete vertex set in that is maximal is a clique et such that ) same, 6 A path (or chain) from to is a sequence for all of distinct vertices ( The path is a cycle if the end points are allowed to be the If there is a path from to we say that are connected The sets disconnected if are not connected for all 6 are, The connected component of a vertex 6 is the set of all vertices connected with A graph is connected if all the pairs of vertices are connected The set from if it is an 6 is an separator if all paths from to intersect The set separator for every connected component of are in two distinct connected components of 6 separates, is a separator of if two vertices in the same or, equivalently, if is disconnected In addition, is a minimal separator of if is a separator no proper subset of separates the graph Unless otherwise stated, the separators we work with will be complete A tree is a connected graph with no cycles In a tree, there is a unique path between any two vertices The vertex is called terminal in a tree if there is only one edge linking with the remaining vertices Definition A1 The partition ) separator of In this case ) of 6 is said to form a decomposition of if is a minimal decomposes into the components %) 22! %) The decom

24 ) position is proper if are not empty Definition A2 The graph is decomposable if it is complete or if there exists a proper decomposition into decomposable graphs %) Assume that is decomposable let )! %) be the set of cliques of Since is decomposable, it is possible to order the vertex sets in ) in a perfect sequence (Blair Barry, W 1993) If we denote P " M, it follows that, for every, ) is a decomposition of %) (auritzen, 1996) We let ) 2 3 be the set of separators of the graph associated with ) et X) ) be a tree defined on the set of cliques of the decomposable graph Definition A3 (The Star Property) Take et, with 1) 9 ) let 2 M for some ) 1) be the two subtrees obtained by removing the edge ) Consider the vertex sets from " D T D The tree is said to have the Star Property for if, for every edge ), ) (A1)! is a decomposition of Blair Barry (1993) show that it is always possible to construct a tree that has the Star Property In addition, they show that such for some edge ) )!2 M K ) is a minimal separator of if only if M ) ) The set of separators associated with will be given by By removing a terminal clique from such a tree, the Star Property is preserved as shown in the next result 23

25 emma A4 et ) ) be a tree defined on the set of cliques of a decomposable graph Assume that has the Star Property for et be a terminal clique in let clique in ) such that ) obtained by removing from Then defined by the set of cliques ) We consider Proof Consider an arbitrary edge ) ) ) be the two subtrees obtained by removing the edge ) et We can assume that subtrees ) Since is terminal in, we have separator for, is also a separator for, it follows that ) Star Property for ) the vertex sets defined in Equation A1 ) ) be the the unique to be the tree is a tree with the Star Property for the decomposable graph If we were to remove the edge ) ) X) D, hence Moreover, ) ) from, with As before, we let from The vertex set associated with UO The vertex set! will be a decomposition of, we would obtain the BM is which is a! is a decomposition of From Therefore the tree has the 24

26 ist of Figures 1 A decomposable graph with four vertices two cliques 26 2 A decomposable graph with eleven vertices four cliques 2 25

27 Figure 1 A decomposable graph with four vertices two cliques 26

28 Figure 2 A decomposable graph with eleven vertices four cliques 2