Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Orientable convexity, geodetic and hull numbers in graphs   We prove three results conjectured or stated by Chartrand, Fink and Zhang
[European J. Combin {\bf 21} (2000) 181--189, Disc. Appl. Math. {\bf 116}
(2002) 115--126, and pre-print of ``The hull number of an oriented graph''].
For a digraph $D$, Chartrand et al. defined the geodetic, hull and convexity
number -- $g(D)$, $h(D)$ and $con(D)$, respectively. For an undirected graph
$G$, $g^{-}(G)$ and $g^{+}(G)$ are the minimum and maximum geodetic numbers
over all orientations of $G$, and similarly for $h^{-}(G)$, $h^{+}(G)$,
$con^{-}(G)$ and $con^{+}(G)$. Chartrand and Zhang gave a proof that $g^{-}(G)
< g^{+}(G)$ for any connected graph with at least three vertices. We plug a gap
in their proof, allowing us also to establish their conjecture that $h^{-}(G) <
h^{+}(G)$.
  If $v$ is an end-vertex, then in any orientation of $G$, $v$ is either a
source or a sink. It is easy to see that graphs without end-vertices can be
oriented to have no source or sink; we show that, in fact, we can avoid all
extreme vertices. This proves another conjecture of Chartrand et al., that
$con^{-}(G) < con^{+}(G)$ iff $G$ has no end-vertices.

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The categories of this paper are: bayes-an
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The categories of this paper are: plasm-ph
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The categories of this paper are: funct-an
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The categories of this paper are: ao-sci
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The categories of this paper are: eess
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The categories of this paper are: cond-mat
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The categories of this paper are: alg-geom
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The categories of this paper are: q-fin
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The categories of this paper are: gr-qc
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The categories of this paper are: nlin
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The categories of this paper are: cs
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The categories of this paper are: mtrl-th
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The categories of this paper are: dg-ga
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The categories of this paper are: chao-dyn
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The categories of this paper are: math-ph
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The categories of this paper are: q-alg