Nordhaus–Gaddum for treewidth

Joret, Gwenaël; Wood, David R.

doi:10.1016/j.ejc.2011.10.005

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%

Sphere Representations, Stacked Polytopes, and the Colin de Verdière Number of a Graph

Mitchell

Yengulalp

2016

Electron. J. Combin.

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We prove that a $k$-tree can be viewed as a subgraph of a special type of $(k+1)$-tree that corresponds to a stacked polytope and that these "stacked'' $(k+1)$-trees admit representations by orthogonal spheres in $\mathbb{R}^{k+1}$. As a result, we derive lower bounds for Colin de Verdière's $\mu$ of complements of partial $k$-trees and prove that $\mu(G) + \mu(\overline{G}) \geq |G| - 2$ for all chordal $G$.

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Section: Introductionmentioning

confidence: 99%

Sphere Representations, Stacked Polytopes, and the Colin de Verdière Number of a Graph

Mitchell

Yengulalp

2016

Electron. J. Combin.

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“…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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Multi-part Nordhaus-Gaddum type problems for tree-width, Colin de Verdière type parameters, and Hadwiger number

Hogben,

Lin,

Young

2016

Preprint

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A traditional Nordhaus-Gaddum problem for a graph parameter β is to find a (tight) upper or lower bound on the sum or product of β(G) and β(G) (where G denotes the complement of G). An r-decomposition G 1 , . . . , G r of the complete graph K n is a partition of the edges of K n among r spanning subgraphs G 1 , . . . , G r . A traditional Nordhaus-Gaddum problem can be viewed as the special case for r = 2 of a more general r-part sum or product Nordhaus-Gaddum type problem. We determine the values of the r-part sum and product upper bounds asymptotically as n goes to infinity for the parameters tree-width and its variants largeur d'arborescence, pathwidth, and proper path-width. We also establish ranges for the lower bounds for these parameters, and ranges for the upper and lower bounds of the r-part Nordhaus-Gaddum type problems for the parameters Hadwiger number, the Colin de Verdière number µ that is used to characterize planarity, and its variants ν and ξ.

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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%

Nordhaus–Gaddum bounds for locating domination

Hernando

Mora

Pelayo

2014

European Journal of Combinatorics

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A dominating set S of graph G is called metric-locating-dominating if it is also locating, that is, if every vertex v is uniquely determined by its vector of distances to the vertices in S . If moreover, every vertex v not in S is also uniquely determined by the set of neighbors of v belonging to S , then it is said to be locating-dominating. Locating, metric-locating-dominating and locatingdominating sets of minimum cardinality are called β-codes, η-codes and λ-codes, respectively. 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. In this paper, we present some Nordhaus-Gaddum bounds for the location number β, the metric-location-domination number η and the location-domination number λ. Moreover, in each case, the graph family attaining the corresponding bound is fully characterized.

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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.

Cited by 6 publications

References 15 publications

Sphere Representations, Stacked Polytopes, and the Colin de Verdière Number of a Graph

Sphere Representations, Stacked Polytopes, and the Colin de Verdière Number of a Graph

Multi-part Nordhaus-Gaddum type problems for tree-width, Colin de Verdière type parameters, and Hadwiger number

Nordhaus–Gaddum bounds for locating domination

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