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

Abstract

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.