Graph Classes: A Survey (Monographs on Discrete Mathematics by Andreas Brandstädt, Van Bang Le, Jeremy P. Spinrad

By Andreas Brandstädt, Van Bang Le, Jeremy P. Spinrad

This well-organized reference is a definitive encyclopedia for the literature on graph periods. It includes a survey of greater than 2 hundred sessions of graphs, geared up by way of varieties of houses used to outline and symbolize the periods, bringing up key theorems and literature references for every. The authors kingdom effects with no evidence, supplying readers with easy accessibility to way more key theorems than are in general present in different mathematical texts. Interconnections among graph sessions also are supplied to make the e-book important to numerous readers.

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5 Generalized perfection There are many generalizations of the parameters \,u,a, and K of a graph; the generalized parameters also satisfy the min-max inequalities. This leads to several concepts containing perfection as a special case. The first such generalization is due to Hell and Roberts [538], and is based on the following notion due to Harary [519] and Sabidussi [939]. 1 Let G and G' be graphs. The lexicographic product GoG' is the graph with vertex set V(G) x V(G') and edge set {(9i,9()(92,92) • 9192 € E(G) V (9l = g2 A && e E(G'))}.

6. 6 Let G be a graph and i be a positive integer. A Ki+\ -free k-coloring of G is a partition of the vertex set of G into k subsets each of which induces a Ki+\-free subgraph of G. Xi(G) denotes the smallest number k for which G has a Ki+\-free k-coloring. (G — G')=u(G)-i. 3 For all positive integers i, all perfect graphs G satisfy Xi(G) = w;(G) and have a min{u>(G),i}-transversal. See also the remarks at the end of the next section.

Note that for every graph G and set S C Ci(G), Ci(S) > Pi(3). This leads to a quantification over all subsets of the set Ki(G). 7 Let i > 2 be an integer. A graph G is AVperfect if for each

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