Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Quasi-efficient domination in grids   Domination of grids has been proved to be a demanding task and with the
addition of independence it becomes more challenging. It is known that no grid
with $m,n \geq 5$ has an efficient dominating set, also called perfect code,
that is, an independent vertex set such that each vertex not in it has exactly
one neighbor in that set. So it is interesting to study the existence of
independent dominating sets for grids that allow at most two neighbors, such
sets are called independent $[1,2]$-sets. In this paper we prove that every
grid has an independent $[1,2]$-set, and we develop a dynamic programming
algorithm using min-plus algebra that computes $i_{[1,2]}(P_m\Box P_n)$, the
minimum cardinality of an independent $[1,2]$-set for the grid graph
$P_m\square P_n$. We calculate $i_{[1,2]}(P_m\Box P_n)$ for $2\leq m\leq 13,
n\geq m$ using this algorithm, meanwhile the parameter for grids with $14\leq
m\leq n$ is obtained through a quasi-regular pattern that, in addition,
provides an independent $[1,2]$-set of minimum size.

7
The categories of this paper are: hep-th
21
The categories of this paper are: adap-org
10
The categories of this paper are: q-bio
14
The categories of this paper are: math-ph
32
The categories of this paper are: atom-ph
31
The categories of this paper are: ao-sci
20
The categories of this paper are: acc-phys
36
The categories of this paper are: cmp-lg
13
The categories of this paper are: quant-ph
27
The categories of this paper are: comp-gas
35
The categories of this paper are: mtrl-th
11
The categories of this paper are: hep-lat
18
The categories of this paper are: econ
23
The categories of this paper are: solv-int
4
The categories of this paper are: cond-mat
9
The categories of this paper are: nlin