在一個有向圖$D$中，對任意兩頂點$u,v$，一條$u$-$v$測地線是每條$u$到$v$的最短有向路徑。
令$I_{D}[u,v]$為一點子集蒐集所有在每條$u$-$v$測地線及$v$-$u$測地線上的頂點。且對$S$為圖$D$的點子集，
令$I_{D}[S]$為所有$u,vin S$的$I_{D}[u,v]$的聯集。
若$I_{D}[S]=S$則稱$S$為一凸子集。令$[S]_{D}$為包含$S$的最小凸子集。若$S$為集合大小最小且滿足$I_{D}[S]$是$D$的頂點集，
則此集合大小即為測地線集合數，記為$g(D)$。若$S$為集合大小最小且滿足$[S]_{D}$是$D$的頂點集，
則此集合大小即為莢集數，記為$h(D)$。對一個連通圖$G$，觀察圖$G$的所有定向圖的$g(D)$及$h(D)$，
取最大的$g(D)$為$g^{+}(G)$，最小的$g(D)$為$g^{-}(G)$；取最大的$h(D)$為$h^{+}(G)$，最小的$h(D)$為$h^{-}(G)$。
我們得到對任意一個連通圖$G$且$G$的頂點集數至少為3，則$g^{+}(G)>g^{-}(G)$且$h^{+}(G)>h^{-}(G)$。
接著我們證得$h^{+}(G)=h^{-}(G)+1$若且唯若$G$同構於$K_{3}$或$K_{1,r}$且$rgeq
2$。我們也得到對每一個連通圖$G$，$h^{+}(G)geq
5$若且唯若$G$的頂點集數至少為5且$G$不同構於$C_{5}$。令$Sh^{*}(G)={h(D):D$為一$n$頂點強連通定向圖$}$。
則我們獲得$Sh^{*}(K_{n})={2}$。令圖$C(n,t)$的頂點集合為${1,2,...,n,x,y}$且邊集合為
${i(i+1):i=1,2,...,n-1}cup {1n}cup {1x}cup {ty}$。
我們也得到若$ngeq 5,t
eq frac{n}{2}$且$3leq tleq n-1$
則$h^{-}(C(n,t))<g^{-}(C(n,t))<h^{+}(C(n,t))=g^{+}(C(n,t))$；
這個結果回答了Farrugia的其中一個問題。

For every pair of vertices $u,v$ in an oriented graph, a $u$-$v$
$geodesic$ is a shortest directed path from $u$ to $v$. For an
oriented graph $D$, let $I_{D}[u,v]$ denoted the set of all
vertices lying on a $u$-$v$ geodesic or a $v$-$u$ geodesic. And
for $Ssubseteq V(D)$, let $I_{D}[S]$ denoted the union of all
$I_{D}[u,v]$ for all $u,vin S$. If $S$ is a $convex$ set then
$I_{D}[S]=S$. Let $[S]_{D}$ denoted the smallest convex set
containing $S$. The $geodetic$ $number$ $g(D)$ of an oriented
graph $D$ is the minimum cardinality of a set $S$ with
$I_{D}[S]=V(D)$. The $hull$ $number$ $h(D)$ of an oriented graph
$D$ is the minimum cardinality of a set $S$ with $[S]_{D}=V(D)$.
For a connected graph $G$, let $g^{-}(G)=$min${g(D)$:$D$ is an
orientation of $G$ $}$ and $g^{+}(G)=$max${g(D)$:$D$ is an
orientation of $G$ $}$. And let $h^{-}(G)=$min${h(D)$:$D$ is an
orientation of $G$ $}$ and $h^{+}(G)=$max${h(D)$:$D$ is an
orientation of $G$ $}$. We show that $h^{+}(G)>h^{-}(G)$ and
$g^{+}(G)>g^{-}(G)$ for every connected graph $G$ with
$|V(G)|geq 3$. Then we show that $h^{+}(G)=h^{-}(G)+1$ if and
only if $G$ is isomorphic to $K_{3}$ or $K_{1,r}$ for $rgeq 2$
and prove that for every connected graph $G$, $h^{+}(G)geq 5$ if
and only if $|V(G)|geq 5$ and $G
cong C_{5}$. Let
$Sh^{*}(G)={h(D)$:$D$ is a strongly connected orientation of $G$
$}$ and we have $Sh^{*}(K_{n})={2}$. Let a graph $C(n,t)$ with
$V(C(n,t))={1,2,...,n,x,y}$ and
$E(C(n,t))={i(i+1):i=1,2,...,n-1}cup {1n}cup {1x}cup
{ty}$. We also have $h^{-} (C(n,t))<g^{-}(C(n,t))<h^{+}(C(n,t))
=g^{+}(C(n,t))$ if $n geq 5$, $t
eq frac{n}{2}$ and $3leq
tleq n-1$. The last result answers a problem of Farrugia in [7].

{1}Introduction}---{1}
{2}Previous results}---{6}
{3}The main results}---{9}
{4}Further research}---{28}

G. Chartrand, J.F. Fink, P. Zhang, $Orientation hull
numbers of a graph$, Congressus Numerantium 161(2003), 181-194.

G. Chartrand, J.F. Fink, P. Zhang, $The hull number of
an oriented graph$, Int. J. Math. Math. Sci. 36(2003),
2265-2275.

G. Chartrand, F. Harary, P. Zhang, $On the hull number ofa graph$, Ars Combin. 57(2000), 129-138.

G. Chartrand, L. Lesniak, $Graphs and Digraphs$, Chapman and Hall, 1996.

G.J. Chang, L.D. Tong, H.T. Wang, $Geodetic specrtum of
graphs$, European Journal of Combinatorics 25(2004), 383-391.

M.G. Evereet, S.B. Seidman, $The hull number of a graph$, Discrete Math. 57(1985), 217-223.

Alastair Farrugia, $Orientation convexity, geodetic and
hull numbers in graphs$, Discrete applied Math. 148(2005), 256-262.

F. Harary, J. Nieminen, $Convexity in graphs$ , J. Diff. Geom. 16(1981), 185-190.

D.B. West, $Introduction to Graph Theorey$, Prentice Hall, 2001.