G是一個簡單圖。對於圖G中的任意兩個點u 和v，我們稱一條u 到v 的最短路徑叫作是一條u-v的測地線。令I_G [u,v]是一個收集圖G中所有u-v測地線上的點的點子集。假設S 為圖G 的點子集，當所有在S 中的任意點u 和v所形成的I_G [u,v]的聯集為圖G 的點集時，我們將S 稱作是圖G 的一個測地線集，並取S 集合個數最小的量為圖G 的測地數。
令F 是圖G 的點子集。在圖G 中，一個關於F 的斯氏樹是包含F 中所有點的一個邊數最少的無圈連通子圖。令S_G [F] 是一個收集圖G中所有落在某一個關於F 的斯氏樹上的點的點子集。當S_G [F]為圖G的點集時，我們將F 稱作是圖G的一個斯氏集，並取F 集合個數最小的量為圖G的斯氏數。

在[9]參考文章中有此一猜想：對於整數r≥3和a≥b≥3 ，存在連通圖G滿足圖G的直徑和半徑皆為r，圖G的測地數為a且其斯氏數為b。此篇論文裡，我們研究此猜想並考慮圖的直徑為半徑加一的情況。

我們證明了以下的結果：

(1) 存在點的個數為4k+9，k≥2的圖G滿足其直徑、半徑皆為4 ，測地數為3+k ，且斯氏數為 3。
(2) 存在點的個數為4k+25，k≥1的圖G滿足其直徑、半徑皆為5 ，測地數為4+k ，且斯氏數為 3。
(3) 存在點的個數為6k+12，k≥1的圖G滿足其直徑為6 ，半徑為5，測地數為2+k ，且斯氏數為 3。
(4) 存在點的個數為9k+15，k≥1的圖G滿足其直徑為7，半徑為6 ，測地數為3+k ，且斯氏數為 3。
(5) 存在點的個數為9k+12，k≥1的圖G滿足其直徑為6 ，半徑為5，測地數為2+2k ，且斯氏數為 3。

關鍵字：測地數；斯氏數；直徑；半徑

A u − v geodesic in G is a shortest path between u and v in G. The geodetic interval I_G[u,v] is the set of all vertices which are lying on some u − v geodesic. A geodetic set S of G is a subset of vertices in G such that ∪_ u,v∈S I_G[u,v] = V (G). The geodetic number is the minimum cardinality of a geodetic set in G, denoted by g(G). Let F be a subset of the vertex set of G. A Steiner tree of F in G is a minimum acyclic connected subgraph of G containing F. The Steiner interval of F in G is the collection of vertices lying on some Steiner trees of F in G, denoted by S_G[F]. A Steiner set is the subset F of vertices in G which satisfies S_G[F] = V (G). We call that F is a Steiner set of G. The Steiner number s(G) of G is the minimum cardinality of a Steiner set of G.

In [9], there is a conjecture: For integers r ≥ 3 and a ≥ b ≥ 3, there exists a connected graph G with rad(G) = diam(G) = r, s(G) = b, and g(G) = a. We study the conjecture and consider the constraint diam(G) = rad(G)+1. In this thesis, we prove the following results:

(1) There exists a graph G of order 4k + 9 with k ≥ 2, diam(G) = rad(G) = 4, g(G) = 3 + k, and s(G) = 3.
(2) There exists a graph G of order 4k + 25 with k ≥ 1, diam(G) = rad(G) = 5, g(G) = 4 + k, and s(G) = 3.
(3) There exists a graph G of order 6k + 12 with k ≥ 1, diam(G) = rad(G) + 1 = 6, g(G) = 2 + k, and s(G) = 3.
(4) There exists a graph G of order 9k + 15 with k ≥ 1, diam(G) = rad(G) + 1 = 7, g(G) = 3 + k, and s(G) = 3.
(5) There exists a graph G of order 9k + 12 with k ≥ 1, diam(G) = rad(G) + 1 = 6, g(G) = 2 + 2k, and s(G) =3.

Keywords: geodetic number; Steiner number; diameter; radius

論文審定書 i
誌謝 ii
中文摘要 iii
Abstract iv
Contents v
List of Figures vi
1 Introduction 1
2 The main results 5
2.1 Previous results and preparations . . . . . . . . . . . . . . . . 5
2.2 The main results to diam(G)=rad(G) . . . . . . . . . . . . . . 10
2.3 The main results to diam(G)=rad(G)+1 . . . . . . . . . . . . 13
3 Conclusion 17

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