Bi-Arc Digraphs and Conservative Polymorphisms

TL;DR

This paper characterizes bi-arc digraphs, a broad class generalizing interval graphs, and provides polynomial algorithms for recognizing them, linking their structure to certain polymorphisms relevant in constraint satisfaction problems.

Contribution

It offers a forbidden obstruction characterization and a polynomial recognition algorithm for bi-arc digraphs, connecting them to conservative polymorphisms and solving open recognition complexity problems.

Findings

Polynomial recognition algorithm for bi-arc digraphs.

Complete dichotomy classification for recognition problems in relational structures.

Bi-arc digraphs generalize interval graphs and related classes.

Abstract

In this paper we study the class of bi-arc digraphs, important from two seemingly unrelated perspectives. On the one hand, they are precisely the digraphs that admit certain polymorphisms of interest in the study of constraint satisfaction problems; on the other hand, they are a very broad generalization of interval graphs. Bi-arc digraphs is the class of digraphs that admit conservative semilattice polymorphisms. There is much interest in understanding structures that admit particular types of polymorphisms, and especially in their recognition algorithms. (Such problems are referred to as metaproblems.) Surprisingly, the class of bi-arc digraphs also describes the class of digraphs that admit certain other kinds of conservative polymorphisms. Thus solving the recognition problem for bi-arc digraphs solves the metaproblem for digraphs for several types of conservative polymorphisms. The complexity of the recognition problem for digraphs with conservative semilattice polymorphisms was an open problem, while it was known to be NP-complete for certain more complex relational structures. We complement our result by providing a complete dichotomy classification of which general relational structures have polynomial or NP-complete recognition problems for the existence of conservative semilattice polymorphisms. Bi-arc digraphs also generalizes the class of interval graphs; in fact it reduces to the class of interval graphs for symmetric and reflexive digraphs. It is much broader than interval graphs and includes other generalizations of interval graphs such as co-threshold tolerance graphs and adjusted interval digraphs. Yet, it is still a reasonable extension of interval graphs, in the sense that it keeps much of the appeal of interval graphs. Our main result is a forbidden obstruction characterization of, and a polynomial recognition for, the class of bi-arc digraphs.