Abstract |
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]. |