# On the thinness and proper thinness of a graph

## TL;DR

This paper introduces the concept of proper thinness as a generalization of thinness in graphs, studies their properties, and explores algorithmic implications for problems on these graph classes.

## Contribution

It defines proper thinness, analyzes its properties, and extends polynomial-time solvability results to broader classes of problems on these graphs.

## Key findings

- Proper thinness generalizes proper interval graphs.
- Certain problems are solvable in polynomial time on graphs with bounded thinness.
- Extended polynomial-time algorithms for problems on graphs with bounded proper thinness.

## Abstract

Graphs with bounded thinness were defined in 2007 as a generalization of interval graphs. In this paper we introduce the concept of proper thinness, such that graphs with bounded proper thinness generalize proper interval graphs. We study the complexity of problems related to the computation of these parameters, describe the behavior of the thinness and proper thinness under three graph operations, and relate thinness and proper thinness with other graph invariants in the literature. Finally, we describe a wide family of problems that can be solved in polynomial time for graphs with bounded thinness, generalizing for example list matrix partition problems with bounded size matrix, and enlarge this family of problems for graphs with bounded proper thinness, including domination problems.