Thinness of product graphs

TL;DR

This paper investigates the thinness parameter of graph products, demonstrating that for most types, the thinness of the product is bounded by a function of the factors' thinness, with some cases lacking such bounds.

Contribution

It provides a comprehensive analysis of how thinness behaves under various graph product operations, extending understanding of this width parameter.

Findings

Thinness of graph products is generally bounded by a function of the factors' thinness.

Most graph products have predictable thinness behavior, often related to sum or product of individual thinness.

Some graph products do not have a bounded thinness function, indicating complex interactions.

Abstract

The thinness of a graph is a width parameter that generalizes some properties of interval graphs, which are exactly the graphs of thinness one. Many NP-complete problems can be solved in polynomial time for graphs with bounded thinness, given a suitable representation of the graph. In this paper we study the thinness and its variations of graph products. We show that the thinness behaves "well" in general for products, in the sense that for most of the graph products defined in the literature, the thinness of the product of two graphs is bounded by a function (typically product or sum) of their thinness, or of the thinness of one of them and the size of the other. We also show for some cases the non-existence of such a function.