T. Kloks and D. Kratsch. Listing all minimal separators of a graph. SIAM Journal on Computing, 27 (1998) 605-613.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....by equating the closed sets of the lattice with the set of minimal separators of an underlying graph. The notion of minimal separator, introduced by Dirac in 1961 to characterize chordal graphs (see [10] has been studied extensively during the past decade on non chordal graphs (see [17] [16], 21] 3] 27] and has yielded many new theoretical and algorithmical graph results. We will apply some of these results to analyzing and decomposing a binary relation and the associated concept lattice. Because of space restrictions, we will mostly limit ourselves to presenting our basic ....

....by a binary relation (see [11] 5] 20] both when one wants to store all the closed sets, and when one simply wants to encounter all of them at least once. 9 In parallel, recent work has been done to generate all the minimal separators or all the minimal xy separators of a graph (see [25] [16], 7] 24] As an illustration of the use that can be made of our new paradigm, we will show how we can easily match the complexity of generating and storing the closed sets obtained by [20] which is O(n ) per closed set, by using the work of [24] who claims a complexity of O(n ) time ....

T. Kloks and D. Kratsch. Listing all minimal separators of a graph. SIAM Journal on Computing, 27:605613, 1998.

....= fa; d; e; f; 3; 4; 5; 6g = S, which shows that S is indeed a minimal separator of GR . c 3 2 4 5 6 f e d a b 1 S C 1 C 2 Figure 3. Separator S = fa; d; e; f; 3; 4; 5; 6g of GR . This enables us to use existing algorithms for generating the minimal separators of a graph (see [21] 20] [15], 5] to efficiently generate the concepts, matching the best complexities of [16] and [9] Moreover, if we add in GR the edges necessary to make S into a clique, defining a new relation R , which is obtained from R by deleting the corresponding crosses, then concept lattice L(R ) is the ....

T. Kloks and D. Kratsch. Listing all minimal separators of a graph. SIAM Journal on Computing, 27:605--613, 1998.

....graph. This is algorithmically interesting because, in the past decade, much research has been done on using minimal separators to eciently solve various graph problems such as chordal embedding ( 19] 2] and in particular several papers deal with the ecient enumeration of minimal separators ([14], 21] 20] 3] 4] pointed out that, using the underlying co bipartite graph and these recent results on the emerging theory of minimal separation, the current best algorithms for generating concepts could easily be matched both in terms of time and space. In this paper, we use graph ....

T. Kloks and D. Kratsch. Listing all minimal separators of a graph. SIAM Journal on Computing, 27:605-613, 1998.

....lattices, by equating the concepts of the lattice with the set of minimal separators of an underlying graph. The notion of minimal separator, introduced by Dirac in 1961 to characterize chordal graphs ( 11] has been extensively studied during the past decade on non chordal graphs (see [22] [21], 28] 3] 38] and has yielded many new theoretical and algorithmical graph results. LIMOS FRE CNRS 2239, Ensemble Scienti que des Czeaux, Universit Blaise Pascal, 63170 Aubire, France. E mail: berry isima.fr, sigayret isima.fr We will apply some of these results to analyzing and ....

.... may want to compute the concepts along with their structure, i.e. the arcs of the graph representing the relationships between elements of the lattice (see [10] In parallel, recent work has been done to generate all the minimal separators or all the minimal xy separators of a graph (see [36] [21], 8] 35] As an illustration of the use that can be made of our new paradigm, we will show how we can easily match the complexity of generating and storing the concepts obtained by [27] which is O(n ) per concept, by using the work of Shen ( 35] who claims a complexity of O(n ) time ....

T. Kloks and D. Kratsch. Listing all minimal separators of a graph. SIAM Journal on Computing, 27:605613, 1998.

....Potential Maximal Cliques ( 9] it also indirectly improves computing the TREEWIDTH and MINIMUM FILL IN on a graph with a polynomial number of minimal separators. Regarding the speci c problem of computing the set of minimal separators of a graph, this has given rise to recent research ( 11] [12], 17] 11] proposed a process with a global complexity of O(n 6 ) per separator, which was later streamlined in ( 12] to O(n 5 ) a complexity also independently obtained by [17] All three papers use the following technique: 1. Choose an arbitrary pair fa; bg of non adjacent vertices, ....

....a polynomial number of minimal separators. Regarding the speci c problem of computing the set of minimal separators of a graph, this has given rise to recent research ( 11] 12] 17] 11] proposed a process with a global complexity of O(n 6 ) per separator, which was later streamlined in ([12]) to O(n 5 ) a complexity also independently obtained by [17] All three papers use the following technique: 1. Choose an arbitrary pair fa; bg of non adjacent vertices, and compute the set of minimal ab separators. 2. Repeat this process on every possible pair fx; yg of non adjacent ....

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T. Kloks and D. Kratsch, \Listing all minimal separators of a graph", SIAM Journal on Computing 27 (1998) 605-613.

....2 mj G j 2 ) time. Proof. Let us analyze the complexity of the algorithm. The sets of vertex sets, like G and G , will be represented by trees, in such manner that the adjunction of a new element and testing that a vertex set belongs to our set will be done in linear time (see for example [20]) We also know by corollary 2 that a call of the function IS PMC takes O(nm) time. We start with the cost of one execution of the function ONE MORE V ERTEX . The cost of the rst for loop is at most j 0 G jnm. But we can strongly reduce this complexity, using a di erent test for verifying ....

T. Kloks and D. Kratsch. Listing all minimal separators of a graph. SIAM J. Comput., 27(3):605613, 1998.

....in the size of the input, i.e. in n and p. 5 Application to some classes of graphs Several classes of graphs have few minimal separators, in the sense that the number of minimal separators of such graphs is polynomially bounded in the size of the graph. Moreover, an algorithm given in [20] computes all the minimal separators of these graphs. For some of these classes of graphs we also have algorithms computing the treewidth and the minimum ll in in polynomial time, using the minimal separators (cf. 19, 8, 3, 31, 23, 14, 25] Di erent proofs have been given for each of these ....

T. Kloks and D. Kratsch. Listing all minimal separators of a graph. SIAM J. Comput., 27(3):605613, 1998.

....all graphs whose minimal separators have size bounded by a fixed integer. Note that every class C can be recognized in polynomial time since every graph of C has at most polynomial many separators and all minimal separators of a graph can be listed in polynomial time per separator [17]. In Figure 1 we show some examples of such graph classes. PSfrag replacements (a) b) c) d) S1 S2 Sr G1 G2 Gr Gr 1 F i F j u v x i y i z i x j y j z j :x i :z i :z j Figure1. a) split graphs whose isolated vertices have bounded degree, b) tree like graphs whose nodes ....

T. Kloks and D. Kratsch. Listing all minimal separators of a graph. SIAM J. Comput., 27(3):605--613, 1998.

....bounded number of minimal separators has a polynomially bounded number of Potential Maximal Cliques, a conjecture also related to the structural aspects of the global set of minimal separators of a graph. Recent research has been done to compute the set of minimal separators of a graph ( 9] [10], 14] 9] proposed a process with a global complexity of O(n 6 ) per separator, which was later streamlined in ( 10] to O(n 5 ) a complexity also independently obtained by [14] All three papers use the following technique: 1. Choose an arbitrary pair fa; bg of non adjacent vertices, and ....

....related to the structural aspects of the global set of minimal separators of a graph. Recent research has been done to compute the set of minimal separators of a graph ( 9] 10] 14] 9] proposed a process with a global complexity of O(n 6 ) per separator, which was later streamlined in ([10]) to O(n 5 ) a complexity also independently obtained by [14] All three papers use the following technique: 1. Choose an arbitrary pair fa; bg of non adjacent vertices, and compute the set of minimal ab separators. 2. Repeat this process on every possible pair fx; yg of non adjacent vertices, ....

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T. Kloks and D. Kratsch. Listing all the minimal separators of a graph. SIAM Journal on Computing, 27:605--613, 1998.

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T. Kloks and D. Kratsch. Listing all minimal separators of a graph. SIAM Journal on Computing, 27 (1998) 605-613.

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T. Kloks and D. Kratsch. Listing all minimal separators of a graph. SIAM J. Comput. 27 (1998) 605613.

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T. Kloks and D. Kratsch. Listing all minimal separators of a graph. SIAM Journal on Computing, 27:605-613, 1998.

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