Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the maximum number of edges in plane graph with fixed exterior face
  degree   A well known Euler's formula consequence's corollary in graph theory states
that: For a connected simple planar graph with $n$ vertices and $m$ edges, and
girth $g$, we have $m \leq \frac{g}{g-2}(n-2)$. We show that a connected simple
plane graph with $n$ vertices and girth $g$, and exterior face of degree $h$
has at most $\frac{g}{g-2}(n-2)- \frac{1}{g-2}(h-g)$ edges. A \emph{convex hull
$g$-angulation} is a connected plane graph in which the exterior face is a
simple $h$-cycle and all inner faces are $g$-cycles. For a given set $S$ of $n$
point in the plane having $h$ points in the boundary of its convex hull, we
present the necessary and sufficient condition to obtain a convex hull
$g$-angulation on $S$. We also determine the number of edges and inner faces in
the convex hull $g$-angulation.

21
The categories of this paper are: adap-org
15
The categories of this paper are: nucl-ex
8
The categories of this paper are: hep-ex
27
The categories of this paper are: comp-gas
0
The categories of this paper are: hep-ph
30
The categories of this paper are: q-alg
29
The categories of this paper are: funct-an
17
The categories of this paper are: q-fin
9
The categories of this paper are: nlin
23
The categories of this paper are: solv-int
20
The categories of this paper are: acc-phys
16
The categories of this paper are: stat
35
The categories of this paper are: mtrl-th
28
The categories of this paper are: alg-geom
32
The categories of this paper are: atom-ph
7
The categories of this paper are: hep-th