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| Title: | 圖之和弦圖數與樹寬
The Chordality and Treewidth of a Graph |
| Authors: | 游朝凱 |
| Contributors: | 張宜武
游朝凱 |
| Keywords: | 和弦圖數
樹寬
樹分解
系列平行圖
Chordality
Treewidth
Tree Decomposition
Series Parallel Graph |
| Date: | 2011 |
| Issue Date: | 2016-05-10 19:33:38 (UTC+8) |
| Abstract: | 對於任何一個圖G = (V;E) ，如果我們可以找到最少的k 個弦圖(V;Ei)，使得E = E1 \\ \\ Ek ，則我們定義此圖G = (V;E) 的chordality為k ；而一個圖G = (V;E) 的樹寬則被定義為此圖所有的樹分解的寬的最小值。在這篇論文中，最主要的結論是所有圖的chordality 會小於或等於它的樹寬；更特別的是，有一些平面圖的chordality 為3，而所有系列平行圖的chordality 頂多為2。
The chordality of a graph G = (V;E) is dened as the minimum k such that we can write E = E1 \\ \\ Ek, where each (V;Ei) is a chordal graph. The treewidth of a graph G = (V;E) is dened to be the minimum width over all tree decompositions of G. In this thesis, the principal result is that the chordality of a graph is at most its treewidth. In particular, there are planar graphs with chordality 3, and series-parallel graphs have chordality at most 2.
Abstract ii
中文摘要iii
1 Introduction 1
1.1 History of Chordality and Treewidth 1
1.2 The De nition of Chordality and Treewidth 2
2 The Chordality of a Graph 6
2.1 Theorems and Examples of Chordality 6
2.2 The Counter Example 14
3 The Treewidth of a Graph 15
3.1 The Treewidth of Some Classes of Graphs 15
3.2 The Chordality of K2;2;2 22
4 Chordality vs. Treewidth 23
4.1 The Weaker Inequality between Chordality and Treewidth 23
4.2 Chordality Bounded by Its Treewidth 24
4.3 The Chordality of Series{Parallel Graphs 37
References 38 |
| Reference: | [1] L.W. Beineke and R.E. Pippert, Properties and characterizations of k-trees, Mathematika, 18 (1971), 141-151.
[2] P. Bumeman, A characterization of rigid circuit graphs, Discrete Mathematics, 9 (1974), 205-212.
[3] M.B. Cozzens and F.S. Roberts, On dimensional properties of graphs, Graphs and Combinatorics, 5 (1989), 29-46.
[4] G.A. Dirac, A property of 4-chromatic graphs and some remarks on critical graphs, J. London Math. Soc., 27 (1952), 85-92.
[5] R.J. Dun, Topology of series parallel-networks, J. Math. Anal. Appl., 10 (1965), 303-318.
[6] F. Gavril, The intersection graphs of subtrees in trees are exactly the chordal graphs, Journal of Combinatorial Theory (B), 16 (1974), 47-56.
[7] Pinar Heggernes, Treewidth, partial k-trees, and chordal graphs, Delpensum INF 334- Institutt for informatikk, (2006).
[8] Terry A. McKee and Edward R. Sceinerman, On the Chordality of a Graph, Journal of Graph Theory, 17 (1993), 221-232.
[9] H.P. Patil, On the structure of k{trees, J. Combin. Inform. System. Sci., 11 (1986), 57-64.
[10] N. Roberston and P.D. Seymour, Graph minors II: algorithmic aspects of tree width, J. of Algorithms, 7 (1986), 309-322.
[11] D.J. Rose, On simple characterizations of k-trees, Discrete Math., 7 (1974), 317-322.
[12] J.R. Walter, Representations of chordal graphs as subtrees of a tree, J. Graph Theory, 2 (1978), 265-267. |
| Description: | 碩士
國立政治大學
應用數學系數學教學碩士在職專班
98972004 |
| Source URI: | http://thesis.lib.nccu.edu.tw/record/#G0098972004 |
| Data Type: | thesis |
| Appears in Collections: | [應用數學系] 學位論文
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