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URN etd-0127111-064740 Author Pei-Lan Yen Author's Email Address No Public. Statistics This thesis had been viewed 5094 times. Download 643 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

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etd-0127111-064740.pdf Date of Submission 2011-01-27