Locally definable vertex set properties are efficiently enumerable

Abstract

We propose a general framework that allows for the study of enumeration of vertex set properties in graphs. We prove that when such a property is locally definable with respect to some order on the set of vertices, then it can be enumerated with linear delay. Our method consists in reducing the considered enumeration problem to the enumeration of paths in directed acyclic graphs. We then apply this general method to enumerate minimal connected dominating sets and maximal irredundant sets in interval graphs and in permutation graphs, as well as maximal irredundant sets in circular-arc graphs and in circular permutation graphs, with linear delay.