Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the Concentration of the Domination Number of the Random Graph   In this paper we study the behaviour of the domination number of the
Erd\H{o}s-R\'enyi random graph $\mathcal{G}(n,p)$. Extending a result of
Wieland and Godbole we show that the domination number of $\mathcal{G}(n,p)$ is
equal to one of two values asymptotically almost surely whenever $p \gg
\frac{\ln^2n}{\sqrt{n}}$. The explicit values are exactly at the first moment
threshold, that is where the expected number of dominating sets starts to tend
to infinity. For small $p$ we also provide various non-concentration results
which indicate why some sort of lower bound on the probability $p$ is necessary
in our first theorem. Concentration, though not on a constant length interval,
is proven for every $p\gg 1/n$. These results show that unlike in the case of
$p \gg \frac{\ln^2n}{\sqrt{n}}$ where concentration of the domination number
happens around the first moment threshold, for $p = O(\ln n/n)$ it does so
around the median. In particular, in this range the two are far apart from each
other.

12
The categories of this paper are: nucl-th
25
The categories of this paper are: patt-sol
26
The categories of this paper are: bayes-an
37
The categories of this paper are: supr-con
11
The categories of this paper are: hep-lat
4
The categories of this paper are: cond-mat
22
The categories of this paper are: chao-dyn
10
The categories of this paper are: q-bio
31
The categories of this paper are: ao-sci
32
The categories of this paper are: atom-ph
27
The categories of this paper are: comp-gas
19
The categories of this paper are: eess
21
The categories of this paper are: adap-org
23
The categories of this paper are: solv-int
16
The categories of this paper are: stat
36
The categories of this paper are: cmp-lg