On finite width questionable representations of orders

TL;DR

This paper investigates finite width questionable representations of orders, exploring their properties, computational implications, and generalizations, including applications to graph classes with bounded tree-width or clique-width.

Contribution

It introduces the concept of balanced tree-questionable-width and establishes relationships between this measure and existing graph width parameters.

Findings

Countable total orders have binary questionable representations.

Polynomial-time algorithms for testing isomorphism and counting linear extensions on certain classes.

Bounded balanced tree-questionable-width does not imply bounded tree-width or clique-width.

Abstract

In this article, we study "questionable representations" of (partial or total) orders, introduced in our previous article "A class of orders with linear? time sorting algorithm". (Later, we consider arbitrary binary functional/relational structures instead of orders.) A "question" is the first difference between two sequences (with ordinal index) of elements of orders/sets. In finite width "questionable representations" of an order O, comparison can be solved by looking at the "question" that compares elements of a finite order O'. A corollary of a theorem by Cantor (1895)is that all countable total orders have a binary (width 2) questionable representation. We find new classes of orders on which testing isomorphism or counting the number of linear extensions can be done in polynomial time. We also present a generalization of questionable-width, called balanced tree-questionable-width, and show that if a class of binary structures has bounded tree-width or clique-width, then it has bounded balanced tree-questionable-width. But there are classes of graphs of bounded balanced tree-questionable-width and unbounded tree-width or clique-width.