C. Berge and P. Duchet, Strongly perfect graphs, Anals of Discrete Math 21, (1984), 57-61.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....C 5 free graphs whose 1 overlap graph is bipartite. First we note, that all three of these classes contain the C 8 and its complement. Therefore the classes O 3 , O 2 and O 1 are neither contained in the class of strict quasi parity graphs [19] nor in the class BIP [4] nor strongly perfect [2]. Two vertices are called an even pair if all induced paths connecting these two vertices have even length. A graph G is called quasi parity if every induced subgraph or its complement contains an even pair. Meyniel [19] proved that quasi parity graphs are perfect. The graph of Figure 7a) shows ....

C.Berge, P.Duchet, Strongly perfect graphs, Ann. Discrete Math. 21 (1984), 57--61

....set in H meeting every maximal clique of H. Note that a graph is perfect if and only if, for every induced subgraph H, there is an independent set meeting every maximum size clique, so a strongly perfect graph is certainly perfect. Strongly perfect graphs were introduced by Berge and Duchet [2] in 1984, and further investigated in papers of Chv atal [9] Ho ang, Maffray and Preissman [14] and Ho ang [13] Attention has principally focussed on relations between the class of strongly perfect graphs and other classes of perfect graphs. For a partial order P = X; the comparability ....

C.Berge and P.Duchet, Strongly perfect graphs, in Topics on Perfect Graphs (C.Berge and V.Chv'atal eds.), North Holland, Amsterdam, 1984, 57--61. 32

....if the vertices of every induced subgraph H of G can be coloured with (H) colours, where (H) is the maximum clique size in H. It can be easily seen that a graph G is perfect if and only if every induced subgraph H contains a stable set intersecting every maximum clique in H. Berge and Duchet [1] de ned a graph G to be strongly perfect if every induced subgraph H of G contains a stable set that intersects every maximal clique in H (as usual, maximal is meant with respect to set inclusion) Chv atal [2] shown that perfectly orderable graphs are strongly perfect and that the greedy ....

C. Berge and P. Duchet, "Strongly Perfect Graphs", in: C. Berge and V. Chvatal, eds., Topics on Perfect Graphs, Annals of Discrete Math. 21 (North Holland, Amsterdam, 1984) 57-61.

....each of whose subgraphs has a predominating node, are perfect. A stable set of a graph G has been termed strong if it has a non empty intersection with all maximal cliques of G. Since no minimally imperfect graph contains a strong stable set by Theorem 2. 2, strongly perfect graphs, introduced in [2] to be graphs all of whose subgraphs admit a strong stable set, are perfect. The notion of strong stable sets was generalized by Hammer and Maffray in [9] They called a node set V 0 V of G = V; E) absorbant if every node in V V 0 has at least one neighbor in V 0 , and then a graph ....

C. Berge and P. Duchet, Strongly Perfect Graphs. In: Topics on Perfect Graphs. (C. Berge and V. Chvatal, eds.), North Holland, Amsterdam (1984) 57-61

....3 for such classes of perfect graphs which minimally imperfect subgraphs may occur after deleting (adding) a critical (anticritical) edge. The large abundance of classes of perfect graphs led us to mention only results for some classical classes: Strongly perfect graphs have been introduced in [2] to be graphs all of whose subgraphs G 0 G admit a stable set that has a non empty intersection with all maximal cliques of G 0 . Weakly triangulated graphs are de ned to have neither holes C k nor antiholes C k with k 5 as induced subgraphs [5] Meyniel [12] called a graph G strict quasi ....

C. Berge and P. Duchet, Strongly Perfect Graphs. In: Topics on Perfect Graphs. (C. Berge and V. Chvatal, eds.), North Holland, Amsterdam (1984) 57-61

....We call a graph perfect if for each induced subgraph, the chromatic number and the maximum clique size coincide. It is an open problem to recognize perfect graphs in polynomial time. It is even an open problem to characterize perfect graphs by forbidden induced subgraphs. Conjecture [1] see also [2]) A graph is perfect iff it has no cycle of odd length greater three and not the complement of a cycle of odd length greater three as an induced subgraph. We call a graph Berge perfect if it has not a cycle and not the complement of a cycle of odd length greater three as an induced subgraph. 8 ....

C. Berge, P. Duchet, Strongly Perfect Graphs, in "Topics on Perfect Graphs" (C. Berge, V. Chvatal ed.), Annals of Discrete Mathematics 21 (1984), pp. 221-224.

....interest in perfectly contractile graphs has been piqued by the results discussed above. We close this section with some suggestions for future research directions. To begin we mention a few classical classes of perfect graphs which may be perfectly contractile. A graph is strongly perfect [4] if each of its induced subgraphs contains a stable set meeting all maximal cliques. It is easy to see that a graph is perfect if and only if each of its induced subgraphs contains a stable set meeting all maximum cliques. Thus, strongly perfect graphs are perfect. A graph in alternately ....

C. Berge, P. Duchet, Strongly perfect graphs, In C. Berge and V. Chv'atal, editors, Ann. Discrete Math., 21 (1984) 57--62.

....1 b 2 ; a 2 b 1 62 E. case 1.1 b 1 b 2 2 E. then fv; a 1 ; b 1 ; b 2 ; a 2 g = C 5 . case 1.2 b 1 b 2 62 E. then fv; a 1 ; b 1 ; x; b 2 ; a 2 g = C 6 . case 2 a 1 b 2 2 E for example. then fv; a 1 ; b 2 ; x; u 2 g = C 5 . 2. 3 Comparison with other classes of perfect graphs Berge and Duchet [3] call a graph G strongly perfect if every induced subgraph has a stable set which meets every maximal cliques. An even pair is a pair of (non adjacent) vertices such that every chordless path which joins the two vertices has an even number of edges. A graph G is said to be strict quasiparity if ....

C. Berge and P. Duchet, strongly perfect graphs, in "Topics on perfect graphs" (C. Berge and V. Chv'atal, Eds.), pp 57-61, North-Holland, Amsterdam, 1984.

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C. Berge and P. Duchet, Strongly perfect graphs, Anals of Discrete Math 21, (1984), 57-61.

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C. Berge and P. Duchet, Strongly perfect graphs, In Topics on Perfect Graphs, C. Berge and V. Chv'atal, editors, Ann. Discrete Math. 21 (1984) 57--62, North Holland, Amsterdam.

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