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

Abstract

We consider the performance of the Depth First Search (DFS) algorithm on the random graph G n, 1+ǫ n , ǫ > 0 a small constant. Recently, Enriquez, Faraud and Ménard [2] proved that the stack U of the DFS follows a specific scaling limit, reaching the maximal height of (1 + o ǫ (1)) ǫ 2 n. Here we provide a simple analysis for the typical length of a maximum path discovered by the DFS.

Depth-First Search Performance in a Random Digraph with Geometric Outdegree Distribution

“…Remark 1.1 Some related results for DFS in an undirected Erdős-Rényi graph G(n, λ/n) are proved by Faraud and Ménard [9] and Diskin and Krivelevich [8], and DFS in a random Erdős-Rényi digraph has been studied for example in the proof of [17,Theorem 3]. These models are closely related to our model with a Poisson outdegree distribution P; they will therefore be further discussed in [12].…”

An improved lower bound on the length of the longest cycle in random graphs

“…This path, in the regime c = 1 + where is a sufficiently small constant has size (1 + o(1)) 2 n w.h.p. This result was extended to = ω(n − 1 3 +o(1) ) by Diskin and Krivelevich [11]. Earlier Krivelevich and Sudakov showed that the DFS algorithm finds a path of length Θ( 2 n) for = ω(n −1/3 log 1/3 n) w.h.p.…”