Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: k-Metric Antidimension: a Privacy Measure for Social Graphs   Let $G = (V, E)$ be a simple connected graph and $S = \{w_1, \cdots, w_t\}
\subseteq V$ an ordered subset of vertices. The metric representation of a
vertex $u\in V$ with respect to $S$ is the $t$-vector $r(u|S) = (d_G(u, w_1),
\cdots, d_G(u, w_t))$, where $d_G(u, v)$ represents the length of a shortest
$u-v$ path in $G$. The set $S$ is called a resolving set for $G$ if $r(u|S) =
r(v|S)$ implies $u = v$ for every $u, v \in V$. The smallest cardinality of a
resolving set is the metric dimension of $G$. In this article we propose, to
the best of our knowledge, a new problem in Graph Theory that resembles to the
aforementioned metric dimension problem. We call $S$ a $k$-antiresolving set if
$k$ is the largest positive integer such that for every vertex $v \in V-S$
there exist other $k-1$ different vertices $v_1, \cdots, v_{k-1} \in V-S$ with
$r(v|S) = r(v_1|S) = \cdots = r(v_{k-1}|S)$, \emph{i.e.}, $v$ and $v_1, \cdots,
v_{k-1}$ have the same metric representation with respect to $S$. The
$k$-metric antidimension of $G$ is the minimum cardinality among all the
$k$-antiresolving sets for $G$.
  In this article, we introduce a novel privacy measure, named $(k,
\ell)$-anonymity and based on the $k$-metric antidimension problem, aimed at
evaluating the resistance of social graphs to active attacks. We, therefore,
propose a true-biased algorithm for computing the $k$-metric antidimension of
random graphs. The success rate of our algorithm, according to empirical
results, is above $80 \%$ and $90 \%$ when looking for a $k$-antiresolving
basis and a $k$-antiresolving set respectively. We also investigate theoretical
properties of the $k$-antiresolving sets and the $k$-metric antidimension of
graphs. In particular, we focus on paths, cycles, complete bipartite graphs and
trees.

3
The categories of this paper are: physics
25
The categories of this paper are: patt-sol
29
The categories of this paper are: funct-an
30
The categories of this paper are: q-alg
26
The categories of this paper are: bayes-an
11
The categories of this paper are: hep-lat
6
The categories of this paper are: astro-ph
28
The categories of this paper are: alg-geom
15
The categories of this paper are: nucl-ex
13
The categories of this paper are: quant-ph
33
The categories of this paper are: plasm-ph
0
The categories of this paper are: hep-ph
35
The categories of this paper are: mtrl-th
31
The categories of this paper are: ao-sci
12
The categories of this paper are: nucl-th
22
The categories of this paper are: chao-dyn