2012
DOI: 10.1016/j.ejc.2011.10.005
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Nordhaus–Gaddum for treewidth

Abstract: We prove that for every n-vertex graph G, the treewidth of G plus the treewidth of the complement of G is at least n − 2. This bound is tight.

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Cited by 6 publications
(6 citation statements)
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“…Inequalities that bound p(G) + p(G) or p(G)p(G) for some parameter p are also called Nordhaus-Gaddum type for the authors of early results on the chromatic number of graphs [22]. Recent examples include tw(G) + tw(G) |G| − 2 [7,12], where tw(G) is the tree-width of G, and upper bounds involving a multiple of |G| for Colin de Verdière type parameters [2]. Leslie Hogben has written a nice survey of Nordhaus-Gaddum problems for Colin de Verdière type parameters, variants of tree-width, and related parameters [10].…”
Section: Introductionmentioning
confidence: 99%
“…Inequalities that bound p(G) + p(G) or p(G)p(G) for some parameter p are also called Nordhaus-Gaddum type for the authors of early results on the chromatic number of graphs [22]. Recent examples include tw(G) + tw(G) |G| − 2 [7,12], where tw(G) is the tree-width of G, and upper bounds involving a multiple of |G| for Colin de Verdière type parameters [2]. Leslie Hogben has written a nice survey of Nordhaus-Gaddum problems for Colin de Verdière type parameters, variants of tree-width, and related parameters [10].…”
Section: Introductionmentioning
confidence: 99%
“…Likewise, tree-width and its variants have played a fundamental role in the theory of graph minors since their (re)introduction by Robertson and Seymour in the early 1980s. Surprisingly, Nordhaus-Gaddum theory (r = 2) has only recently been studied for tree-width and its variants, with the sum lower bound established in [7,13] and the sum upper bound established in [13]. In addition to other uses in graph theory that motivated their introduction, Colin de Verdière type parameters have played an important role in the study of minimum rank/maximum nullity of real symmetric matrices described by a graph (see [3,8,12]).…”
Section: Introductionmentioning
confidence: 99%
“…Theorem 1.1 is established by Theorems 2.1, 2.2, 2.4, and Corollary 2.5. It should be noted that Theorem 1.1(a) was established for r = 2 and tree-width in [13] and better results are known for Theorem 1.1(b) in the case that r = 2: Kostochka [14] showed that for n ≥ 5, η(2; n) = 6 5 n , so lim n→∞ η(2;n) n…”
Section: Introductionmentioning
confidence: 99%
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“…A Nordhaus-Gaddum bound is a tight lower or upper bound on the sum or product of a parameter of a graph G and its complement G [1,9,12]. For example, in [5] it was shown that for any graph G of order n, γ(G) + γ(G) ≤ n + 1, the equality being true only if {G, G} = {K n , K n }.…”
Section: Introductionmentioning
confidence: 99%