Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Cycle convexity and the tunnel number of links   In this work, we introduce a new graph convexity, that we call Cycle
Convexity, motivated by related notions in Knot Theory.
  For a graph $G=(V,E)$, define the interval function in the Cycle Convexity as
$I_{cc}(S) = S\cup \{v\in V(G)\mid \text{there is a cycle }C\text{ in }G\text{
such that } V(C)\setminus S=\{v\}\}$, for every $S\subseteq V(G)$. We say that
$S\subseteq V(G)$ is convex if $I_{cc}(S)=S$. The convex hull of $S\subseteq
V(G)$, denoted by $Hull(S)$, is the inclusion-wise minimal convex set $S'$ such
that $S\subseteq S'$. A set $S\subseteq V(G)$ is called a hull set if
$Hull(S)=V(G)$. The hull number of $G$ in the cycle convexity, denoted by
$hn_{cc}(G)$, is the cardinality of a smallest hull set of $G$.
  We first present the motivation for introducing such convexity and the study
of its related hull number. Then, we prove that: the hull number of a 4-regular
planar graph is at most half of its vertices; computing the hull number of a
planar graph is an $NP$-complete problem; computing the hull humber of chordal
graphs, $P_4$-sparse graphs and grids can be done in polynomial time.

25
The categories of this paper are: patt-sol
6
The categories of this paper are: astro-ph
17
The categories of this paper are: q-fin
32
The categories of this paper are: atom-ph
29
The categories of this paper are: funct-an
19
The categories of this paper are: eess
15
The categories of this paper are: nucl-ex
28
The categories of this paper are: alg-geom
13
The categories of this paper are: quant-ph
21
The categories of this paper are: adap-org
10
The categories of this paper are: q-bio
37
The categories of this paper are: supr-con
33
The categories of this paper are: plasm-ph
34
The categories of this paper are: chem-ph
27
The categories of this paper are: comp-gas
24
The categories of this paper are: dg-ga