Algorithmic Traversals of Infinite Graphs

TL;DR

This paper extends the theory of graph traversals, including BFS and DFS, to infinite graphs, establishing their properties, bounds, and minimality in a wellordered context.

Contribution

It generalizes search algorithms to infinite graphs, characterizes their traversals, and provides bounds on order type modifications.

Findings

Extended search theory to infinite graphs.

Proved bounds on order type modifications.

Characterized traversals as lexicographically minimal.

Abstract

A traversal of a connected graph is a linear ordering of its vertices all of whose initial segments induce connected subgraphs. Traversals, and their refinements such as breadth-first and depth-first traversals, are computed by various graph searching algorithms. We extend the theory of generic search and breadth-first search from finite graphs to wellordered infinite graphs, recovering the notion of "search trees" in this context. We also prove tight upper bounds on the extent to which graph search and breadth-first search can modify the order type of the original graph, as well as characterize the traversals computed by these algorithms as lexicographically minimal.