Well-quasi-ordering and finite distinguishing number

TL;DR

This paper characterizes the relationship between the distinguishing number and well-quasi-ordering in hereditary graph classes, showing that classes above Bell numbers with finite distinguishing number contain complex structures, and provides bounds on decision procedures.

Contribution

It proves that hereditary classes above Bell numbers with finite distinguishing number contain boundary classes for well-quasi-ordering, completing the classification of such classes.

Findings

Hereditary classes above Bell numbers with finite distinguishing number contain infinite antichains.

All hereditary classes below Bell numbers are well-quasi-ordered.

Decision procedure runtime is bounded by a quadruple exponential function.

Abstract

Balogh, Bollobas and Weinreich showed that a parameter that has since been termed the distinguishing number can be used to identify a jump in the possible speeds of hereditary classes of graphs at the sequence of Bell numbers. We prove that every hereditary class that lies above the Bell numbers and has finite distinguishing number contains a boundary class for well-quasi-ordering. This means that any such hereditary class which in addition is defined by finitely many minimal forbidden induced subgraphs must contain an infinite antichain. As all hereditary classes below the Bell numbers are well-quasi-ordered, our results complete the answer to the question of well-quasi-ordering for hereditary classes with finite distinguishing number. We also show that the decision procedure of Atminas, Collins, Foniok and Lozin to decide the Bell number (and which now also decides well-quasi-ordering for classes of finite distinguishing number) has run time bounded by an explicit (quadruple exponential) function of the order of the largest minimal forbidden induced subgraph of the class.