L. Lov asz, Normal Hypergraphs and the Weak Perfect Graph Conjecture, Discrete Math. 2 (1972) pp. 253-267  Home/Search   Document Not in Database   Summary   Related Articles   Check

....if G e is imperfect. Analogously, we call an edge e not contained in a perfect graph G anticritical if G e is imperfect. A perfect graph G without isolated nodes is critically perfect if G has only critical edges. The complement of a critically perfect graph is again perfect (due to Lov asz [9]) and has the property that adding an edge not contained in the graph so far yields an imperfect graph. We call the complements of critically perfect graphs anticritically perfect. We look for bicritically perfect graphs which are both critically and anticritically perfect: the deletion and ....

....v and v , respectively, with N(v) v = N(v ) v. That means, L(F e ) arises from L(F ) by replacing the node v in L(F ) by the clique v; v . Analogously, L(F e can be generated from L(F ) by replacing the node v in L(F ) by the stable set v; v . Due to the Replacement Lemma [9], L(F e ) and L(F e are perfect since L(F ) is, hence vv cannot be a critical edge of L(F e ) Theorem 4 implies that F e not an H graph. 2. Thus, none of the 10 A graphs with 10 edges is an H graph. This implies: Corollary 7 There is no bicritically perfect graph on 10 ....

L. Lov asz, Normal Hypergraphs and the Weak Perfect Graph Conjecture, Discrete Math. 2 (1972) pp. 253-267

....as many cliques to cover all nodes of G 0 as a maximum stable set of G 0 has nodes) Since complementation transforms stable sets into cliques and colorings into clique coverings, we have (G) G) and (G) G) where G denotes the complement of G. Hence, Berge conjectured and Lov asz [8] proved that a graph G is perfect if and only if its complement G is. The second Berge 1 conjecture concerns a characterization of perfect graphs via forbidden subgraphs. It is a simple observation that chordless odd cycles C 2k 1 with k 2, termed odd holes, and their complements C 2k 1 , ....

....Conjecture reads: odd holes and odd antiholes are the only minimally imperfect graphs. In order to give a characterization of minimally imperfect graphs (and thereby to verify or falsify the Strong Perfect Graph Conjecture) many fascinating structures of such graphs have been discovered, see [8, 11]. The motivation to ask for measuring imperfectness was the following: we studied in [14, 15, 16] critical edges in perfect graphs G, i.e. edges e the deletion of which yields an imperfect graph G e. In order to decide whether the resulting graph G e is still almost perfect or already very ....

L. Lovasz, Normal Hypergraphs and the Weak Perfect Graph Conjecture, Discrete Math. 2 (1972) 253-267.

....in polynomial time for perfect graphs, see [8] the structure of perfect graphs is not wellunderstood. In particular, the SPGC still seems to be out of reach. On the other hand, the investigation of minimally imperfect graphs has revealed that these graphs have quite strong properties, see e.g. [5, 9, 13, 14, 15, 16, 18, 19]. That motivated us to introduce a new class of extremal cases with respect to perfectness: critically perfect graphs. We provide several examples and prove some basic properties in section 2. We are interested in relating the class CP of critically perfect graphs to well known classes of perfect ....

....4 and maximal degree Delta(G) n Gamma 3. Proof. Let G = V; E) be a critically perfect graph. i) Suppose e 2 E is not contained in a triangle. Then e is neither H critical nor A critical. Hence, there is G e G s.t. G e Gamma e is a monster. Lov asz s characterization of perfect graphs [13] says ff(G e ) G e ) n but ff(G e Gamma e) G e Gamma e) n. Therefore, G e Gamma e) G e ) note that (G e ) G e Gamma e) and ff(G e ) ff(G e Gamma e) holds and e is contained in the intersection of all maximum cliques of G e . It follows (G) 2, since e is a maximal clique ....

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L. Lov' asz, Normal Hypergraphs and the Weak Perfect Graph Conjecture. Discrete Math. 2 (1972) 253-267

....0 ) 8G 0 G. Since complementation transforms stable sets into cliques and colorings into clique coverings, we have (G) G) and (G) G) where G denotes the complement of G. Hence, Berge conjectured that a graph G is perfect if and only if its complement G is. This was proven by Lov asz [16] and is nowadays known as Perfect Graph Theorem. The second Berge conjecture concerns a characterization of perfect graphs via forbidden subgraphs. It is a simple observation that chordless odd cycles C 2k 1 with k 2, termed odd holes, and their complements C 2k 1 , called odd antiholes, are ....

....graphs (and thereby to verify or falsify the Strong Perfect Graph Conjecture) many strong structural properties of such graphs have been discovered. e.g. every minimally imperfect graph G admits an extraordinary symmetry with respect to its maximum cliques and stable sets. Theorem 2. 1 (Lov asz [16]) Each minimally imperfect graph G has exactly (G) G) 1 nodes and, for every node x of G, the graph G x can be partitioned into (G) cliques of size (G) and into (G) stable sets of size (G) Theorem 2.2 (Padberg [21] Every minimally imperfect graph G on n nodes has precisely n maximum ....

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L. Lov asz, Normal Hypergraphs and the Weak Perfect Graph Conjecture. Discrete Math. 2 (1972) 253-267

....0 has nodes) Since complementation transforms stable sets into cliques and colorings into clique coverings, we have (G) G) and (G) G) where G denotes the complement of G. Hence, Berge conjectured that a graph G is perfect if and only if its complement G is. This was proven by Lov asz [10] and is nowadays known as Perfect Graph Theorem. The second Berge conjecture concerns a characterization of perfect graphs via forbidden subgraphs. It is a simple observation that chordless odd cycles C 2k 1 with k 2, termed odd holes, and their complements C 2k 1 , called odd antiholes, are ....

.... x and y with N(x) y N(y) i.e. all neighbors of x except eventually y belong to the neighborhood of y) G does not admit twins (antitwins) i.e. two nodes x and y such that all remaining nodes of G are adjacent to both or to none of x and y (to either x or to y) due to the Replacement Lemma [10] (Antitwin Lemma [13] Note that the property of being a comparable pair, twins, or antitwins does not depend on whether or not x and y are adjacent. Furthermore, no minimally imperfect graph G contains an even pair (two nodes x and y such that all chordless paths connecting x and y have even ....

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L. Lov asz, Normal Hypergraphs and the Weak Perfect Graph Conjecture. Discrete Math. 2 (1972) 253-267

.... (G) G) for all graphs, as one needs at least (G) colors just to cover the vertices of the maximum clique. Several interesting properties of perfect graphs should be noted. First, the perfect graph theorem of Lov asz indicates that a graph is perfect if and only if its complement is perfect, [Lov72]. This statement is equivalent to saying that for all induced subgraphs G 0 of G, ff(G 0 ) ae(G 0 ) where ff(G 0 ) is the size of the largest stable set in G 0 , and ae(G 0 ) is the size of the smallest number of cliques that cover all vertices of G 0 . Thus, in effect studying ....

L. Lov'asz. Normal hypergraphs and the weak perfect graph conjecture. Discrete Mathematics, 2, 1972.

.... (G) G) for all graphs, as one needs at least (G) colors just to cover the vertices of the maximum clique. Several interesting properties of perfect graphs should be noted. First, the perfect graph theorem of Lov asz indicates that a graph is perfect if and only if its complement is perfect, [40]. This statement is equivalent to saying that for all induced subgraphs G 0 of G, ff(G 0 ) ae(G 0 ) where ff(G 0 ) is the size of the largest stable set in G 0 , and ae(G 0 ) is the size of the smallest number of cliques that cover all vertices of G 0 . Thus, in effect studying ....

L. Lov' asz, Normal hypergraphs and the weak perfect graph conjecture, Discrete Math., 2 (1972), pp. 253--267.

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L. Lov'asz, Normal hypergraphs and the weak perfect graph conjecture, Discrete Math., v.2 N 3, (1972) 253-267.

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