On the Performance of the Depth First Search Algorithm in Supercritical Random Graphs

TL;DR

This paper analyzes the typical length of maximum paths found by the Depth First Search algorithm in supercritical random graphs, building on recent work about the algorithm's stack behavior.

Contribution

It provides a simple analysis of the maximum path length discovered by DFS in supercritical Erdős–Rényi graphs, complementing existing results on the DFS stack height.

Findings

Maximum path length scales with epsilon and n

DFS stack reaches maximal height of approximately epsilon^2 n

Results enhance understanding of DFS performance in supercritical graphs

Abstract

We consider the performance of the Depth First Search (DFS) algorithm on the random graph $G\left(n,\frac{1+\epsilon}{n}\right)$, $\epsilon>0$ a small constant. Recently, Enriquez, Faraud and M\'enard [2] proved that the stack $U$ of the DFS follows a specific scaling limit, reaching the maximal height of $(1+o_{\epsilon}(1))\epsilon^2n$. Here we provide a simple analysis for the typical length of a maximum path discovered by the DFS.