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URN etd-0623106-173155
Author Jung-Ting Hung
Author's Email Address m922040012@student.nsysu.edu.tw
Statistics This thesis had been viewed 5097 times. Download 1474 times.
Department Applied Mathematics
Year 2005
Semester 2
Degree Master
Type of Document
Language English
Title The Hull Numbers of Orientations of Graphs
Date of Defense 2006-06-16
Page Count 37
Keyword
  • hull number
  • 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].
    Advisory Committee
  • none - chair
  • none - co-chair
  • none - co-chair
  • none - co-chair
  • Li-Da Tong - advisor
  • Files
  • etd-0623106-173155.pdf
  • indicate in-campus access immediately and off_campus access in a year
    Date of Submission 2006-06-23

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