Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Changing and unchanging of the domination number of a graph: Path
  addition numbers   Given a graph $G = (V,E)$ and two its distinct vertices $u$ and $v$. The
$(u,v)$-$P_k$-{\em addition graph} of $G$ is the graph $G_{u,v,k-2}$ obtained
from disjoint union of $G$ and a path $P_k: x_0,x_1,..,x_{k-1}$, $k \geq 2$, by
identifying the vertices $u$ and $x_0$, and identifying the vertices $v$ and
$x_{k-1}$. We prove that (a) $ \gamma(G)-1 \leq \gamma(G_{u,v,k})$ for all $k
\geq 1$, and (b) $\gamma(G_{u,v,k}) > \gamma(G)$ when $k \geq 5$. We also
provide necessary and sufficient conditions for the equality $\gamma(G_{u,v,k})
= \gamma(G)$ to be valid for each pair $u,v \in V(G)$. pair $u,v \in V(G)$.

11
The categories of this paper are: hep-lat
7
The categories of this paper are: hep-th
30
The categories of this paper are: q-alg
13
The categories of this paper are: quant-ph
10
The categories of this paper are: q-bio
35
The categories of this paper are: mtrl-th
33
The categories of this paper are: plasm-ph
14
The categories of this paper are: math-ph
27
The categories of this paper are: comp-gas
16
The categories of this paper are: stat
0
The categories of this paper are: hep-ph
1
The categories of this paper are: cs
15
The categories of this paper are: nucl-ex
29
The categories of this paper are: funct-an
20
The categories of this paper are: acc-phys
22
The categories of this paper are: chao-dyn