在這篇論文中,我們先介紹了一個演算法來找出梯形圖上的4-拓展樹,接著介紹了一個在排列圖上改良過後的3-拓展樹, 然後我們依據這個演算法作雛形, 在梯形圖來找出邊數最多為2n 的拓展者, 最後我們提出一個反證證明 區間圖上沒有2-拓展樹. 至於複雜度方面,找出梯形圖的4-拓展樹為 O(n), 在排列圖上改良過後的3-拓展樹為O(n)(原來是O(m + n)) ,在梯形圖找出邊數最多為2n 的拓展者為O(n)

A t-spanner of a graph G is a subgraph H of G, which the distance between any two vertices in H is at most t times their distance in G. A tree t-spanner of G is a t-spanner which is a tree. In this dissertation, we discuss the t-spanners on
trapezoid, permutation, and interval graphs. We first introduce an O(n) algorithm for finding a tree 4-spanner on trapezoid graphs. Then, give an O(n)algorithm for finding a tree 3-spanner on permutation graphs, improving the existed O(n + m)
algorithm. Since the class of permutation graphs is a subclass of trapezoid graphs, we can apply the algorithm on permutation graphs to find the approximation of a tree 3-spanner on trapezoid graphs in O(n) time with edge bound 2n. Finally, we show that not all interval graphs have a tree 2-spanner.

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0
Chapter 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Chapter 2. Permutation, Interval, and Trapezoid Graphs . . . . . . . 4
Chapter 3. Tree 4-Spanners on Trapezoid Graphs . . . . . . . . . . . 9
3.1 Prelimiaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.2 Ekkehard’s Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.3 Finding a Tree 4-spanner of a Trapezoid Graph . . . . . . . . . . . . 12
3.4 The Correctness of Algorithm3.B . . . . . . . . . . . . . . . . . . . . 15
3.5 Implementing Algorithm 3.B in O(n) Time . . . . . . . . . . . . . . . 25
Chapter 4. Tree 3-Spanners on Permutation Graphs . . . . . . . . . . 32
4.1 Madanlal’s Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.2 An O(n) Algorithm for Finding a Tree 3-spanner . . . . . . . . . . . 34
4.3 The Correctness of Algorithm4.B . . . . . . . . . . . . . . . . . . . . 36
Chapter 5. Finding a 3-spanner of Edge Bound 2n on Trapezoid
graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.1 Finding a 3-spanner . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.2 The Correctness of Algorithm5.A . . . . . . . . . . . . . . . . . . . . 43
Page
Chapter 6. Tree t-spanners on Interval Graphs . . . . . . . . . . . . . 45
6.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
6.2 An Example of One Interval Graph Without Having Tree 2-spanner . . . 46
Chapter 7. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

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