Intersection models and forbidden pattern characterizations for 2-thin and proper 2-thin graphs

TL;DR

This paper characterizes 2-thin and proper 2-thin graphs through intersection models, forbidden patterns, and their relation to known graph classes, advancing understanding of their structure and recognition complexity.

Contribution

It provides new characterizations of 2-thin and proper 2-thin graphs as intersection graphs and forbidden patterns, and relates proper thinness to bandwidth.

Findings

Independent 2-thin graphs are interval bigraphs.

Proper independent 2-thin graphs are bipartite permutation graphs.

Upper bounds proper thinness in terms of bandwidth.

Abstract

The \emph{thinness} of a graph is a width parameter that generalizes some properties of interval graphs, which are exactly the graphs of thinness one. Graphs with thinness at most two include, for example, bipartite convex graphs. Many NP-complete problems can be solved in polynomial time for graphs with bounded thinness, given a suitable representation of the graph. \emph{Proper thinness} is defined analogously, generalizing proper interval graphs, and a larger family of NP-complete problems are known to be polynomially solvable for graphs with bounded proper thinness. The complexity of recognizing 2-thin and proper 2-thin graphs is still open. In this work, we present characterizations of 2-thin and proper 2-thin graphs as intersection graphs of rectangles in the plane, as vertex intersection graphs of paths on a grid (VPG graphs), and by forbidden ordered patterns. We also prove that independent 2-thin graphs are exactly the interval bigraphs, and that proper independent 2-thin graphs are exactly the bipartite permutation graphs. Finally, we take a step towards placing the thinness and its variations in the landscape of width parameters, by upper bounding the proper thinness in terms of the bandwidth.