Constrained Planarity and Augmentation Problems

A clustered graph C=(G,T) consists of an undirected graph G and a rooted tree T in which the leaves of T correspond to the vertices of G=(V,E). Each vertex m in T corresponds to a subset of the vertices of the graph called ``cluster''. c-planarity is a natural extension of graph planarity for clustered graphs, and plays an important role in automatic graph drawing. The complexity status of c-planarity testing is unknown. It has been shown by Dahlhaus, Eades, Feng, Cohen that c-planarity can be tested in linear time for c-connected graphs, i.e., graphs in which the cluster induced subgraphs are connected. In the first part of the thesis, we provide a polynomial time algorithms for c-planarity testing of specific planar clustered graphs: Graphs for which - all nodes corresponding to the non-c-connected clusters lie on the same path in T starting at the root of T, or graphs in which for each non-connected cluster its super-cluster and all its siblings in T are connected, - for all clusters m G-G(m) is connected. The algorithms are based on the concepts for the subgraph induced planar connectivity augmentation problem, also presented in this thesis. Furthermore, we give some characterizations of c-planar clustered graphs using minors and dual graphs and introduce a c-planar augmentation method. Parts II deals with edge deletion and bimodal crossing minimization. We prove that the maximum planar subgraph problem remains NP-complete even for non-planar graphs without a minor isomorphic to either K(5) or K(3,3), respectively. Further, we investigate the problem of finding a minimum weighted set of edges whose removal results in a graph without minors that are contractible onto a prespecified set of vertices. Finally, we investigate the problem of drawing a directed graph in two dimensions with a minimal number of crossings such that for every node the incoming and outgoing edges are separated consecutively in the cyclic adjacency lists. It turns out that the planarization method can be adapted such that the number of crossings can be expected to grow only slightly for practical instances

Maximum Planar Subgraph on Graphs not Contractive to K5 or K3,3

The maximum planar subgraph problem is well studied. Recently, it has been shown that the maximum planar subgraph problem is NP-complete for cubic graphs. In this paper we prove shortly that the maximum planar subgraph problem remains NP-complete even for graphs without a minor isomorphic to K5 or K3,3 , respectively

Triangulating Clustered Graphs

A clustered graph C=(G,T) consists of an undirected graph G and a rooted tree T in which the leaves of T correspond to the vertices of G=(V,E) . Each vertex mu in T corresponds to a subset of the vertices of the graph called ''cluster''. C -planarity is a natural extension of graph planarity for clustered graphs. As we triangulate a planar embedded graph so that G is still planar embedded after triangulation, we consider triangulation of a c -connected clustered graph that preserve the c -planar embedding. In this paper, we provide a linear time algorithm for triangulating c -connected c -planar embedded clustered graphs C=(G,T) so that C is still c -planar embedded after triangulation. We assume that every non-trivial cluster in C has at least two childcluster. This is the first time, this problem was investigated

Drawing cycles in networks

In this paper we show how a graph that contains a specified cycle can be drawn in the plane such that the cycle is drawn circularly while the rest of the graph is layouted orthogonally. We also show how to extend this algorithm to deal with a set of disjoint cycles at once

Planarization With Fixed Subgraph Embedding

The visualization of metabolic networks using techniques of graph drawing has recently become an important research area. In order to ease the analysis of these networks, readable layouts are required in which certain known network components are easily recognizable. In general, the topology of the drawings produced by traditional graph drawing algorithms does not reflect the biologists' expert knowledge on particular substructures of the underlying network. To deal with this problem we present a constrained planarization method---an algorithm which computes a graph layout in the plane preserving the predefined shape for the specified substructures while minimizing the overall number of edge-crossings

Planarization With Fixed Subgraph Embedding

The visualization of metabolic networks using techniques of graph drawing has recently become an important research area. In order to ease the analysis of these networks, readable layouts are required in which certain known network components are easily recognizable. In general, the topology of the drawings produced by traditional graph drawing algorithms does not reflect the biologists' expert knowledge on particular substructures of the underlying network. To deal with this problem we present a constrained planarization method---an algorithm which computes a graph layout in the plane preserving the predefined shape for the specified substructures while minimizing the overall number of edge-crossings

Drawing cycles in networks

In this paper we show how a graph that contains a specified cycle can be drawn in the plane such that the cycle is drawn circularly while the rest of the graph is layouted orthogonally. We also show how to extend this algorithm to deal with a set of disjoint cycles at once

Subgraph Induced Connectivity Augmentation

Given a planar graph G=(V,E) and a vertex set Wsubseteq V , the subgraph induced planar connectivity augmentation problem asks for a minimum cardinality set F of additional edges with end vertices in W such that G'=(V,Ecup F) is planar and the subgraph of G' induced by W is connected. The problem arises in automatic graph drawing in the context of c -planarity testing of clustered graphs. We describe a linear time algorithm based on SPQR-trees that tests if a subgraph induced planar connectivity augmentation exists and, if so, constructs an minimum cardinality augmenting edge set

Cupe - the CUBIC Pathway Editor

Cupe(CUBIC Pathway Editor) is a graphical editor for the automatic or interactive generation and display of metabolic networks. Cupe combines the user guidance by its graphical user interface (GUI) with the ability of automatic graph drawing and the possibility for manual interaction. Furthermore, it provides a programming interface for analysis, simulation and cross linking of reactions. One of the outstanding features of Cupe is its automatic layout mechanism which is provided by utilising the well-known AGD library. The adaptation and development of layout algorithms for the requirements of metabolic networks is an interdisciplinary cooperation between the Department of Computer Science, Cologne University, the Chair of Algorithm Engineering, Dortmund University, and the Cologne University Bioinformatics Center. Poster presentation in 14th International Symposium on Graph Drawin

On the Weighted Minimal Deletion of Rooted Bipartite Minors

We investigate the problem of finding a minimal weighted set of edges whose removal results in a graph without minors that are contractible onto a prespecified set of vertices. Such minors are called rooted. The problem of a minimal weighted deletion of all rooted K(i,3)-minors for a fixed integer i is proved to be NP-hard on general graphs. Furthermore, a polynomial time algorithm is developed for the rooted K(1,3)-minor deletion problem on planar graphs while for the rooted K(2,3)-free minor planar graphs a characterization is presented