N. Sbihi, Algorithme de recherche d'un stable de cardinalite maximum dans un graphe sans etoile, Discrete Math. 29 (1980) 53-76  Home/Search   Document Not in Database   Summary   Related Articles   Check

....algorithmic graph problem occuring in many models in Computer Science and Operations Research. It is NPcomplete, even for triangle free graphs ( 22] for planar graphs of degree 3 ( 13] and for (K 1;4 ; K 4 e) free graphs ( 6] whereas it is solvable in polynomial time for claw free graphs ([20, 23]) On the other hand, its complexity is unknown for P 5 free graphs. This recently led to the investigation of a variety of graph classes de ned by forbidding some small subgraphs. For example, in [18] as an application of the so called conic reduction, Lozin gives an O(n ) time solution for ....

N. Sbihi, Algorithme de recherche d'un stable de cardinalite maximum dans un graphe sans etoile, Discrete Math. 29 (1980) 53-76

....this case [5] A graph is perfect if its chromatic number is equal to the size of its maximum clique. F (r) in any instance r of R can be done in PTIME. Proof: The theorem follows from Lemma 11 and the fact that a maximum independent set in a claw free graph can be found in polynomial time [28, 26]. 2 We show now the second approach that directly yields an O(n ) complexity bound. r) in any instance r of R can be done in O(n ) time where n is the number of tuples in r. Proof: Suppose that G d1 ;r = V; E 1 ) and G d2 ;r = V; E 2 ) Then G fd1 ;d 2 g;r = V; E 1 [ E 2 ) By ....

M. Sbihi. Algorithme de Recherche d'un Stable de Cardinalite Maximum dans un Graphe sans  Etoile. Discrete Mathematics, 29:53-76, 1980.

.... graph (i.e. a graph whose every vertex has at most one non adjacent vertex in the opposite part) This characterization has been used to solve in polynomial time the maximum stable set problem in general fork free graphs generalizing an old result for K 1;3 free graphs due to Minty [25] and Sbihi [28] and several particular algorithms for subclasses of fork free graphs [10, 13, 20] This remarkable result stimulated us to search for a generalization of fork free bipartite graphs. This search has been resulted recently in a characterization of the structure of Star 1;2;2 free bipartite graphs ....

....approach to compute a maximum stable set in a graph is the augmenting graph technique. Given a graph G and a stable set S in G, this technique searches for a bipartite subgraph which augments S. This approach has been applied efficiently to solve the problem in the class of K 1;3 free graphs [25, 28]. Recently, based on a characterization of fork free bipartite graphs, Alekseev extended this result to general fork free graphs [1] The same approach has been used by Mosca [26] to develop a polynomial algorithm for the problem in (P 6 ; C 4 ) free graphs. Several new results on this topic can ....

N. Sbihi, Algorithme de recherche d'un stable de cardinalit'e maximum dans un graphe sans 'etoile, Discrete Mathematics 29 (1980) 53-76.

.... Delta Delta Delta A A A Q Q Q Q Q G 7 Figure 4. Minimal forbidden induced subgraphs for (Chair free graphs) Recently Alekseev [1] proved that the stability number problem is polynomially solvable in the class of all Chair free graphs. It generalizes well known results of Minty [13] and Sbihi [16] on claw free graphs. Corollary 3 extends Alekseev s result to a wider class of all (G 1 ; G 2 ; G 7 ) free graphs (see Figure 4) RRR 14 2001 Page 19 Example 5 (P free graphs) Corollary 4 The minimal forbidden induced subgraphs for the substitutional closure of P free graphs are G 1 ....

N. Sbihi, Algorithme de recherche d'un stable de cardinalit'e maximum dans un graphe sans 'etoile, Discrete Math. 29 (1) (1980) 53--76 (in French)

....Q(G) 1; 0; 0; 1; 0) 5. RRR 30 2001 Page 11 The conflict graph H for Q(G) is shown in Figure 2. By Theorem 3, ff(H) 5 with fab; ae; de; bd; bcg being a maximum independent set in H. Note that H is neither perfect nor claw free, i.e. it does not belong to well known ff polynomial classes [3, 7, 8]. Problem 1 Is there a polynomial time algorithm for recognizing 2 path realizable graphs and constructing of some 2 path realization u u u u u H H Phi Phi H H Phi Phi Phi Phi H H Delta Delta A A A A Delta Delta b e a d c G u u u u u u u u u u u u Gamma Gamma Gamma Gamma Gamma ....

N. Sbihi, Algorithme de recherche d'un stable de cardinalit'e maximum dans un graphe sanes 'etoile, Discrete Math. 29 (1980) 53--76 (in French)

....r for which G is K 1;r free satisfies r = ff L 1 . If G is a graph with independent claw centers, then hN(x)i is a claw free graph for any x 2 V (G) otherwise we have two adjacent claw centers) Since the determination of the independence number is polynomial in claw free graphs (see [3] [7]) ff L (G) can be computed in polynomial time for any graph with independent claw centers (although it is NP complete in general) It is easy to see that also the assumption of independent claw centers can be verified in polynomial time. Thus, it is convenient to restate Theorem 1 in the ....

Sbihi, N.: Algorithmes de recherche d'un stable de cardinalite maximum dans un graphe sans etoile. Discrete Math. 29(1980) 53-76.

.... properties of such graphs were observed in [183] 120] and [184] and first results on hamiltonian properties were proved in [85] 50] 148] and [155] However, probably more importantly, were the observations that the determination of the independence number is polynomial (see [151] and [172]) and that the Berge s Perfect Graph Conjecture holds (see [158] in claw free graphs. In general, we follow the most common graph theoretical terminology and notation, and for concepts not defined here we refer to [20] Unless otherwise mentioned, throughout the paper by a graph we always mean a ....

....path, which is, given a matching M , a path whose edges are alternately in M and E(G) Gamma M , and whose end vertices are not incident to any edge of M . When going to the line graph of a graph, a maximum matching is transformed into a maximum independent set. Using this idea Sbihi in [172] gives a polynomial algorithm for finding a maximum independent set in a claw free graph, and Minty, in [151] describes a different but also polynomial algorithm to find an independent set with maximum weight in a weighted claw free graph. 42 Theorem 5.8 [172] 151] There is a polynomial ....

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Sbihi, N.: Algorithmes de recherche d'un stable de cardinalite maximum dans un graphe sans etoile. Discrete Math. 29 (1980) 53-76

....1;3 . The class of K 1;3 free graphs includes all the line graphs and is well studied in many aspects. For instance, it is known that, for K 1;3 free graphs, the strong Berge conjecture holds [1] and certain problems that are NP complete in the general setting are polynomially solvable for them [2, 3]. Since the middle of the 70s the articles have begun to appear that are devoted to study of conditions sufficient for hamiltonicity of a K 1;3 free graph (see Section 3 of the newest survey by Gould [4] It is the following conjecture extending Thomassen s conjecture for line graphs [5] that ....

N. Sbihi, "Algorithme de recherche d'un stable de cardinalit'e maximum dans un graphe sans 'etoile," Discrete Math., 29, No. 1, 53--76 (1980).

....problem have been proposed, e.g. 11, 9, 5] for bipartite graphs and [2, 3, 15, 6] for general graphs. Moreover, these polynomial time algorithms have been extended to those solving more general problems, for instance, the maximum weight cardinality stable set problem for claw free graphs [18, 16, 14], the linear matroid parity problem [12, 13, 7, 17] and the linear delta matroid parity problem [10] In this paper, we will deal with the maximum weight stable set problem for claw free graphs. We call the complete bipartite graph K 1;3 a claw. A graph is said to be claw free if it does not ....

....is one of the forbidden subgraphs of line graphs [1] That is, the line graphs are claw free, and the maximum weight stable set problem for claw free graphs is a generalization of the maximum weight matching problem. Up to date, three polynomial time algorithms, one by Minty [16] one by Sbihi [18] and the third by Lov asz and Plummer [14] have been proposed for the maximum cardinality stable set problem for claw free graphs. Furthermore, it has been believed that Minty s algorithm is the only algorithm which can be extended to the weighted problem. This method finds a maximum weight stable ....

N. Sbihi, Algorithme de recherche d'un stable de cardinalite maximum dans un graphe sans etoile, Discrete Math. 29 (1980), 53-76. 14

....of line graphs and several crucial properties of the matching problem extend to the stable set problem in claw free graphs. So, due to this strong analogy, it is not surprising that there exist polynomially bounded algorithms for finding a maximum (weighted) stable set in a claw free graph ( 13] [17], 12] It is, conversely, very surprising that the nice polyhedral properties of the matching polytope do not extend to the polytope STAB(G) associated with a claw free graph G. On the contrary, as showed by Giles and Trotter [9] when G is a claw free graph, the minimal defining system for ....

N. SBIHI, Algorithme de recherche d'un stable de cardinalite maximum dans un graphe sans etoile, Discrete Mathematics 29 (1980), 53-76.

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N. Sbihi, Algorithme de recherche d'un stable de cardinalite maximum dans un graphe sans etoile, Discrete Math. 29 (1980) 53-76

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N. Sbihi, Algorithme de recherche d'un stable de cardinalite maximum dans un graphe sans etoile, Discrete Math. 29 (1980) 53-76

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N. Sbihi, Algorithme de recherche d'un stable de cardinalit'e maximum dans un graphe sans 'etoile, Discrete Math. 29 (1980) 53--76 (in French)

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