臺灣博碩士論文加值系統

論文提要：
Oellermann 與 Tian 在1990 年[3]提出在樹上找Steiner n-center 的演算法，但是沒有清楚地證明其正確性而且所使用一些引理的證明也已經無法取得，因此我們在論文中做出一個證明以支持該演算法。

他們證明了當n 大於等於2 時，樹的n-center 包含在
n+1-center 中，並提出是否一般圖的n-center 也有此包含關係的問題。葉老師在2004 年[7]舉出一些反例，於是Oellermann 有了這樣的問題：是否有一圖形其n-center 與(n-1)-center 交集為空集合？本文舉出數個包含關係不成立的反例以及無窮多個同類的圖滿足其2-center 與4-center 交集為空集合。

Abstract
Oellermann and Tian presented an algorithm for finding the n-center of a tree in 1990 [3], but the correction of the algorithm seems not sound. In this paper we give a clear proof of the validity of their algorithm.

They also showed a containment relationship for a tree T, C_n(T) is contained in C_{n+1}(T)
for n ≧ 2. We present some graphs G for which C_n(G) is not contained in C_{n+1}(G). Oellermann asked the following question: Can the n-center and (n-1)-center be disjoint? Though the
problem is not solved yet, we present an infinite family of graphs G such that C_2(G)
and C_4(G) are disjoint.

Contents ........................... i
1 Introduction ..................... 1
2 Steiner center in trees .......... 3
3 Examples ........................ 12
References ........................ 17

References
[1] L. W. Beineke, O. R. Oellermann and R. E. Pippert, On the Steiner median of
a tree, Discrete Appl. Math. 68 (1996) 249-258.
[2] G. Chartrand and O. R. Oellermann, S. Tian and H. -B. Zhou, Steiner distance in graphs, Casopis Pest. Mat. 114 (1989) 399-410.
[3] O. R. Oellermann and S. Tian, Steiner centers in graphs, J. Graph Theory, 14 (1990) 585-597.
[4] O. R. Oellermann, Some Open Problems in Graph Theory,
URL: http://www.uwinnipeg.ca/»ooellerm/open problems/index.html
[5] O. R. Oellermann, On Steiner centers and Steiner medians of graphs, Networks, 34 (1999) 258-263.
[6] O. R. Oellermann, Private communication, 1999.
[7] H. G. Yeh, A note on Steiner centers and Steiner medians of graphs, preprint, 2004.