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Long and short paths in uniform random recursive dags. (English) Zbl 1230.60092
The authors’ main concern is the model of the uniform random recursive $$k$$-directed acyclic graph. This random directed graph is constructed as follows. There is a root vertex labeled as 0 and each succesive node labeled from 1 to $$n$$ is joined to $$k$$ of the previous nodes chosen uniformly at random with replacement. The authors provide results about the length of a random path starting from a certain node and leading to the root node. In particular, they show convergence in probability to a certain constant which they provide explicitly. However, the main result of the paper regards the length of the shortest path to the root that starts at vertex $$n$$. In particular, they show that when divided by $$\log n$$, this converges in probability to a certain constant which is determined implicitly through a certain function. Moreover, they show that the maximum shortest distance also satisfies this.

##### MSC:
 60J80 Branching processes (Galton-Watson, birth-and-death, etc.) 05C80 Random graphs (graph-theoretic aspects) 05A15 Exact enumeration problems, generating functions 05C05 Trees
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