On the pre- and post-positional semi-random graph processes

TL;DR

This paper compares two semi-random graph processes, showing they are equally efficient for certain properties like $d$-degeneracy and $k$-connectivity, and identifies properties where their performance differs.

Contribution

It establishes asymptotic equivalence of the two processes for specific graph properties and addresses an open case for $k$-connectivity.

Findings

Both processes are equally fast for constructing graphs with $d$-degenerate subgraphs.

They are also equally efficient for creating $k$-connected graphs, including the previously unresolved case $k=2$.

There exist properties where the two processes differ in construction speed.

Abstract

We study the semi-random graph process, and a variant process recently suggested by Nick Wormald. We show that these two processes are asymptotically equally fast in constructing a semi-random graph $G$ that has property ${\mathcal P}$, for the following examples of ${\mathcal P}$: - ${\mathcal P}$ is the set of graphs containing a $d$-degenerate subgraph, where $d\ge 1$ is fixed; - ${\mathcal P}$ is the set of $k$-connected graphs, where $k\ge 1$ is fixed. In particular, our result of the $k$-connectedness above settles the open case $k=2$ of the original semi-random graph process. We also prove that there exist properties ${\mathcal P}$ where the two semi-random graph processes do not construct a graph in ${\mathcal P}$ asymptotically equally fast. We further propose some conjectures on ${\mathcal P}$ for which the two processes perform differently.