Incremental DFS Trees on Arbitrary Directed Graphs

TL;DR

This paper introduces an algorithm for dynamically maintaining a DFS tree in directed graphs as edges are inserted, with a worst-case time complexity of O(m·n), and establishes lower bounds for related problems.

Contribution

It presents the first algorithm for incremental DFS trees on arbitrary directed graphs with proven worst-case complexity bounds.

Findings

Algorithm processes edge insertions in O(m·n) time

Lower bounds established for related incremental DFS problems

Applicable to dynamic graph analysis and real-time network monitoring

Abstract

We present a new algorithm for maintaining a DFS tree of an arbitrary directed graph under any sequence of edge insertions. Our algorithm requires a total of $O(m\cdot n)$ time in the worst case to process a sequence of edge insertions, where $n$ is the number of vertices in the graph and $m$ is the total number of edges in the final graph. We also prove lower bounds for variations of this problem.