Unary Automatic Graphs: An Algorithmic Perspective

TL;DR

This paper explores algorithmic properties of infinite unary automatic graphs derived from finite graphs, providing efficient polynomial-time algorithms for key connectivity and reachability questions.

Contribution

It introduces polynomial-time algorithms for fundamental problems in unary automatic graphs, improving upon previous non-uniform or non-elementary methods.

Findings

Polynomial-time algorithms for node membership in infinite components

Efficient reachability determination using automata

Improved algorithms over previous non-uniform approaches

Abstract

This paper studies infinite graphs produced from a natural unfolding operation applied to finite graphs. Graphs produced via such operations are of finite degree and automatic over the unary alphabet (that is, they can be described by finite automata over unary alphabet). We investigate algorithmic properties of such unfolded graphs given their finite presentations. In particular, we ask whether a given node belongs to an infinite component, whether two given nodes in the graph are reachable from one another, and whether the graph is connected. We give polynomial-time algorithms for each of these questions. For a fixed input graph, the algorithm for the first question is in constant time and the second question is decided using an automaton that recognizes the reachability relation in a uniform way. Hence, we improve on previous work, in which non-elementary or non-uniform algorithms were found.