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URN etd-0127111-064740
Author Pei-Lan Yen
Author's Email Address No Public.
Statistics This thesis had been viewed 5066 times. Download 635 times.
Department Applied Mathematics
Year 2010
Semester 1
Degree Ph.D.
Type of Document
Language English
Title A study of convexity in directed graphs
Date of Defense 2011-01-19
Page Count 70
Keyword
  • convexity spectrum ratio
  • difference of convexity spectra
  • strong convexity spectrum
  • convexity spectrum
  • convex set
  • orientable convexity number
  • convexity number
  • Abstract Convexity in graphs has been widely discussed in graph theory and G.
    Chartrand et al. introduced the convexity number of oriented graphs
    in 2002. In this thesis, we follow the notions addressed by them and
    develop an extension in some related topics of convexity in directed
    graphs.
    Let D be a connected oriented graph. A set S subseteq V(D)
    is convex in D if, for every pair of vertices x, yin S,
    the vertex set of every x-y geodesic (x-y shortest directed
    path) and y-x geodesic in D is contained in S. The convexity number con(D) of a nontrivial oriented graph D is
    the maximum cardinality of a proper convex set of D. We show that
    for every possible triple n, m, k of integers except for k=4,
    there exists a strongly connected digraph D of order n, size
    m, and con(D)=k.
    Let G be a graph. We define
    the convexity spectrum S_{C}(G)={con(D): D is an
    orientation of G} and the strong convexity spectrum
    S_{SC}(G)={con(D): D is a strongly connected orientation of
    G}. Then S_{SC}(G) ⊆ S_{C}(G). We show that for any
    n ≠ 4, 1 ≤ a ≤ n-2 and a n ≠ 2, there exists a
    2-connected graph G with n vertices such that
    S_C(G)=S_{SC}(G)={a,n-1}, and there is no connected graph G of
    order n ≥ 3 with S_{SC}(G)={n-1}. We also characterizes the
    convexity spectrum and the strong convexity spectrum of complete
    graphs, complete bipartite graphs, and wheel graphs. Those convexity
    spectra are of different kinds.
    Let the difference of convexity spectra D_{CS}(G)=S_{C}(G)-
    S_{SC}(G) and the difference number of convexity spectra
    dcs(G) be the cardinality of D_{CS}(G) for a graph G. We show
    that 0 ≤ dcs(G) ≤ ⌊|V(G)|/2⌋,
    dcs(K_{r,s})=⌊(r+s)/2⌋-2 for 4 ≤ r ≤ s,
    and dcs(W_{1,n-1})= 0 for n ≥ 5.
    The convexity spectrum ratio of a sequence of simple graphs
    G_n of order n is r_C(G_n)= liminflimits_{n to infty}
    frac{|S_{C}(G_n)|}{n-1}. We show that for every even integer t,
    there exists a sequence of graphs G_n such that r_C(G_n)=1/t and
    a formula for r_C(G) in subdivisions of G.
    Advisory Committee
  • Ko-Wei Lih - chair
  • Dah-Jyh Guan - co-chair
  • Gerard Jennhwa Chang - co-chair
  • Xuding Zhu - co-chair
  • Sen-Peng Eu - co-chair
  • Tsai-Lien Wong - co-chair
  • Hong-Gwa Yeh - co-chair
  • Li-Da Tong - advisor
  • Files
  • etd-0127111-064740.pdf
  • indicate access worldwide
    Date of Submission 2011-01-27

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