叢集圖形平面化是藉由添加啞交叉點至叢集圖形的底層圖形，使其轉換成具備叢集平面性質的叢集圖形的一系列操作。平面化的啟發式演算法含兩階段：暫除邊而得到具平面性質的子圖，然後將暫除的邊經平面化操作再補回圖形。其中第二階段一般牽涉到在機件圖形上的路徑搜尋。衡量平面化演算法的準則為產生的啞交叉點個數。

叢集圖形平面化的經典啟發式演算法將暫除邊補回圖形時保持叢集邊界圈。但構造叢集邊界圈會對每個叢集的外延邊的循環次序造成多餘限制，阻礙了潛在的良好嵌入以及其上所能找到的更好的解。

本論文提出針對叢集圖形平面化問題的改良啟發式演算法。構造叢集邊界圈的步驟拆作構造叢集邊界點及叢集邊界邊兩部分，其中叢集邊界邊的存在會對叢集外延邊的循環次序造成多餘限制。因此本論文提出的改良啟發式演算法將構造叢集邊界邊的步驟延後到暫除邊補完之後才實際構造。在此之上，構造並擴充捷徑機件子圖至機件圖上，使得在此擴充機件圖上尋找最短加權路徑即等價於在眾多嵌入中尋找單邊補回的最佳解。

本論文分別從理論及實驗兩方面檢驗此改良演算法。從理論角度驗證，此改良演算法中的單邊補回方法，對於叢集連通的叢集圖形，能夠解得其某類嵌入集下的單邊補回問題的最佳解。而從實驗角度驗證，此改良演算法確實勝過經典演算法，總和來看改良演算法所致生的啞交叉點個數縮減為經典演算法所致生的啞交叉點個數的 89.1% 。

Planarization of clustered graph is a series of operations to transform the underlying graph by creation of degree four crossing dummies such that the result becomes a c-planar clustered graph. The heuristic for planarization consists of two stages: finding a subgraph by discarding some edges, and then reinsert discarded edges back through planarization operations. The edge reinsertion stage generally involves path finding on certain gadgets. The criteria of planarization is to minimize the number of incurred crossing dummies.

The classic method for planarization of clustered graph deal with this problem by starting from a maximal c-planar sub-clustered graph and then repetitively doing single edge insertion while maintaining cluster boundary cycles. But the modeling of cluster boundary cycles put unnecessary constraint on the cyclic ordering of outgoing edges of each cluster, hence prohibit some potentially good embeddings in which better solution can be found.

The thesis proposes an improved heuristic algorithm for planarization of clustered graph. In the thesis, the modeling of cluster boundary cycles breaks into modeling of boundary points and boundary edges, and the modeling of boundary edges are deferred after edge reinsertion stage has been finished. Moreover, shortcut gadgets are augmented to the gadget such that searching a shortest weighted path on the augmented gadget is effectively finding an optimal solution for single edge insertion amongst more embeddings at once.

The proposed method is both theoretically and experimentally examined. Theoretically that the proposed method finds an optimal solution for single edge insertion among a certain type embedding set for a c-connected clustered graph. And experimentally that the proposed method does outperform the classic method in the sense that the overall average of incurred crossings is reduced to about 89.1% in the experiment.

[BDM02] Giuseppe Battista, Walter Didimo, and A. Marcandalli. Planarization of clustered
graphs. In Petra Mutzel, Michael Jünger, and Sebastian Leipert, editors, Graph
Drawing, volume 2265, pages 60–74. Springer Berlin Heidelberg, 2002.
[BETT98] Giuseppe Di Battista, Peter Eades, Roberto Tamassia, and Ioannis G Tollis. Graph
drawing: algorithms for the visualization of graphs. 1998.
[BGL + 97] Giuseppe Di Battista, Ashim Garg, Giuseppe Liotta, Roberto Tamassia, Emanuele
Tassinari, and Francesco Vargiu. An experimental comparison of four graph draw-
ing algorithms. Computational Geometry, 7(5–6):303 – 325, 1997. 11th ACM
Symposium on Computational Geometry.
[BM04] John M. Boyer and Wendy J. Myrvold. On the cutting edge: Simplified o(n) pla-
narity by edge addition. Journal of Graph Algorithms and Applications, 8(3):241–
273, 2004.
[BTT84] C. Batini, M. Talamo, and R. Tamassia. Computer aided layout of entity re-
lationship diagrams. Journal of Systems and Software, 4(2–3):163 – 173, 1984.
Entity-Relationship Approach to Databases and Related Software.
[CDF + 08] Pier Francesco Cortese, Giuseppe Di Battista, Fabrizio Frati, Maurizio Patrignani,
and Maurizio Pizzonia. C-planarity of c-connected clustered graphs. Journal of
Graph Algorithms and Applications, 12(2):225–262, 2008.
[CG12] Markus Chimani and Carsten Gutwenger. Advances in the planarization method:
Effective multiple edge insertions. J. Graph Algorithms Appl., 16(3):729–757, 2012.
[CGMW09] Markus Chimani, Carsten Gutwenger, Petra Mutzel, and Christian Wolf. Inserting
a vertex into a planar graph. In Proceedings of the Twentieth Annual ACM-SIAM
Symposium on Discrete Algorithms, SODA ’09, pages 375–383, Philadelphia, PA,
USA, 2009. Society for Industrial and Applied Mathematics.
[DBT96] G. Di Battista and R. Tamassia. On-line planarity testing. SIAM Journal on
Computing, 25(5):956–997, 1996.
[Dji95] HristoN. Djidjev. A linear algorithm for the maximal planar subgraph problem.
In SelimG. Akl, Frank Dehne, Jörg-Rüdiger Sack, and Nicola Santoro, editors,
Algorithms and Data Structures, volume 955 of Lecture Notes in Computer Science,
pages 369–380. Springer Berlin Heidelberg, 1995.
[FCE95] Qing-Wen Feng, RobertF. Cohen, and Peter Eades. Planarity for clustered graphs.
In Paul Spirakis, editor, Algorithms — ESA ’95, volume 979, pages 213–226.
Springer Berlin Heidelberg, 1995.
[Fen97] Qingwen Feng. Algorithms for drawing clustered graphs, 1997.
[GJ79] Michael R. Garey and David S. Johnson. Computers and Intractability: A Guide
to the Theory of NP-Completeness. W. H. Freeman & Co., New York, NY, USA,
1979.
77[GJ83] M. Garey and D. Johnson. Crossing number is np-complete. SIAM Journal on
Algebraic Discrete Methods, 4(3):312–316, 1983.
[GM01] Carsten Gutwenger and Petra Mutzel. A linear time implementation of spqr-trees.
In Joe Marks, editor, Graph Drawing, volume 1984 of Lecture Notes in Computer
Science, pages 77–90. Springer Berlin Heidelberg, 2001.
[GM03] Carsten Gutwenger and Petra Mutzel. An experimental study of crossing mini-
mization heuristics. In Graph Drawing, pages 13–24, 2003.
[GMW05] Carsten Gutwenger, Petra Mutzel, and René Weiskircher. Inserting an edge into
a planar graph. Algorithmica, 41(4):289–308, 2005.
[Gut10] Carsten Gutwenger. Application of SPQR-trees in the planarization approach for
drawing graphs. PhD thesis, 2010.
[HT73] John Hopcroft and Robert Tarjan. Algorithm 447: Efficient algorithms for graph
manipulation. Commun. ACM, 16(6):372–378, June 1973.
[HT74] John Hopcroft and Robert Tarjan. Efficient planarity testing. J. ACM, 21(4):549–
568, October 1974.
[MGB + 98] Petra Mutzel, Carsten Gutwenger, Ralf Brockenauer, Sergej Fialko, Gunnar W.
Klau, Michael Krüger, Thomas Ziegler, Stefan Näher, David Alberts, Dirk Ambras,
Gunter Koch, Michael Jünger, Christoph Buchheim, and Sebastian Leipert. A
library of algorithms for graph drawing. In Graph Drawing, pages 456–457, 1998.
[MZ99] Petra Mutzel and Thomas Ziegler. The constrained crossing minimization prob-
lem. In Jan Kratochvíyl, editor, Graph Drawing, volume 1731 of Lecture Notes in
Computer Science, pages 175–185. Springer Berlin Heidelberg, 1999.
[OGDF] O-G-D-F. The open graph drawing framework. http://www.ogdf.net .
[Tam14] Roberto Tamassia, editor. Handbook of Graph Drawing and Visualization. CRC
Press, 2014.
[Whi32] Hassler Whitney. Non-separable and planar graphs. Transactions of the American
Mathematical Society, 34(2):339–362, 1932.
[WKWL99] Shih Wei-Kuan and Hsu Wen-Lian. A new planarity test. Theoretical Computer
Science, 223(1–2):179 – 191, 1999.