### Title page for etd-0623106-173155

URN etd-0623106-173155 Jung-Ting Hung m922040012@student.nsysu.edu.tw This thesis had been viewed 5203 times. Download 1511 times. Applied Mathematics 2005 2 Master English The Hull Numbers of Orientations of Graphs 2006-06-16 37 hull number 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 anoriented graph \$D\$, let \$I_{D}[u,v]\$ denoted the set of allvertices lying on a \$u\$-\$v\$ geodesic or a \$v\$-\$u\$ geodesic. Andfor \$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 setcontaining \$S\$. The \$geodetic\$ \$number\$ \$g(D)\$ of an orientedgraph \$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 anorientation of \$G\$ \$}\$ and \$g^{+}(G)=\$max\${g(D)\$:\$D\$ is anorientation of \$G\$ \$}\$. And let \$h^{-}(G)=\$min\${h(D)\$:\$D\$ is anorientation of \$G\$ \$}\$ and \$h^{+}(G)=\$max\${h(D)\$:\$D\$ is anorientation 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 andonly 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\$ ifand only if \$|V(G)|geq 5\$ and \$Gcong 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))

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