圖形的凸集特性在圖論中是一個廣泛討論的問題，而在2002年，G. Chartrand 等人研究了有向圖上的凸集數問題。在此篇論文中，我們延伸他們的概念並且研究在有向圖中的凸集特性相關問題。

對一連通有向圖D中，令S為頂點集V(D)的一子集。若對S內任意兩頂點x,y，每條最短x-y有向路徑及最短y-x有向路徑上的頂點皆落在S內，則稱S為D的一個凸子集。且D的最大凸子集且為真子集的集合大小即為D的凸集數，記為con(D)。對任意的整數n, m, k，除了k ≠ 4，我們架構了一個強連通有向圖D，使得D的點數為n，邊數為m，且凸集數為k。

對於一無向圖G，我們定義G的圖形凸集譜為S_{C}(G)={con(D)| D 為 G 的一個定向圖}，且圖形的強凸集譜為
S_{SC}(G)={con(D)| D 為G 的一個強連通定向圖
}。則我們知道S_{SC}(G) ⊆ S_{C}(G)。我們證明了對於任意整數a, n，當n ≠ 4，1 ≤ a
≤ n-2，且a ≠ 2時，存在一 n 個點的二連通無向圖
G，滿足S_C(G)=S_{SC}(G)={a,n-1}。我們也知道不存在滿足
S_{SC}(G)={n-1} 的連通圖
G。我們分別刻劃了完全圖，輪圖，及完全二佈圖的圖形凸集譜及強凸集譜，
我們發現這三類圖的凸集譜及強凸集譜的特性完全不同。

對於一連通圖G，我們定義其圖形凸集譜差集為D_{CS}(G)=S_{C}(G)-
S_{SC}(G)，其圖形凸集譜差集數dcs(G)
為圖形凸集譜差集的集合大小。我們知道0 ≤ dcs(G) ≤ ⌊
點數/2⌋。對於完全二佈圖，當每個分佈至少包含四個點以
上時，其圖形凸集譜差集數為⌊
點數/2⌋-2。對於點數大於4以上的輪圖，其圖形凸集譜差集數為
0。

對於一系列 n 個點的簡單圖 G_n 來說，我們定義凸集譜比值為
r_C(G_n)= liminflimits_{n to infty} frac{|S_{C}(G_n)|}{n-1}。
我們知道對任意的正偶數 t，存在一系列 n 個點的簡單圖
G_n，使得
r_C(G_n)=1/t。對於連通圖的一延展圖(subdivision)，我們也給了一個公式來計算其凸集譜比值。

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.

誌謝 i
Acknowledgements ii
摘要 iii
Abstract iv
Contents v
List of Figures vi
1 Introduction 1
1.1 Motivation and skeleton . . . . . . . . . . . . . . . . . . . . . 1
1.2 Basic notation . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Convexity number of graphs . . . . . . . . . . . . . . . . . . . 4
1.4 Convexity number of product graphs . . . . . . . . . . . . . . 5
1.5 Ployconvex graphs . . . . . . . . . . . . . . . . . . . . . . . . 10
1.6 H-convex graphs . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.7 Bounds of the convexity number . . . . . . . . . . . . . . . . . 12
1.8 Forcing convexity number of graphs . . . . . . . . . . . . . . . 15
1.9 Computational complexity . . . . . . . . . . . . . . . . . . . . 15
1.10 Convexity number of digraphs . . . . . . . . . . . . . . . . . . 15
1.11 Digraphs with prescribed order and convexity number . . . . . 16
1.12 Orientable convexity numbers . . . . . . . . . . . . . . . . . . 17
2 Convexity spectra of graphs 20
2.1 Constructing graphs with specific convexity spectra . . . . . . 20
2.2 Convexity spectra of complete graphs . . . . . . . . . . . . . . 24
2.3 Constructing strongly connected oriented graphs with fixed
order, size, and convexity number . . . . . . . . . . . . . . . . 28
3 The differences of convexity spectra of graphs 34
3.1 General properties of convexity numbers and bounds for dcs(G) 34
3.2 Complete bipartite graphs K_{r,s} with dcs(K_{r,s}) = ⌊(r+s)/2⌋-2 . . 35
3.3 Wheel graphs W_{1,n−1} with dcs(W_{1,n−1}) = 0 . . . . . . . . . . . 42
3.4 Convexity spectrum ratio . . . . . . . . . . . . . . . . . . . . . 49
Bibliography 53
Index 56

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