A bipartite graph is biplanar if the vertices can be placed on two parallel lines (layers) inthe plane such that there are no edge crossings when edges are drawn as line segments between the layers. In this paper we study the 2-LAYER PLANARIZATION problem: Can k edges be deleted from a given graph G so that the remaining graph is biplanar? This problem is NP-complete, and remains so if the permutation of the vertices in one layer is fixed (the 1-LAYER PLANARIZATION problem). We prove that these problems are fixed-parameter tractable by giving linear-time algorithms for their solution (for fixed k). In particular, we solve the 2-LAYER PLANARIZATION problem in O(k · 6 k +|G|) time and the 1-LAYER PLANARIZATION problem in O(3 k ·|G|) time. We also show that there are polynomial-time constant-approximation algorithms for both problems.

We give a linear-time algorithm to decide whether a graph has a planar LL-drawing, i.e. a planar drawingo two parallel lines. This has previo)L/ beenkno wnoLq fo trees. We utilize this resultto oult planar drawings on three lines for a generalization of bipartite graphs, also in linear time.

An h-layer drawing of a graph G is a planar drawing of G in which each vertex is placed on one of h parallel lines and each edge is drawn as a straight line between its end-vertices. In such a drawing, we say that an edge is proper if its endpoints lie on adjacent layers, flat if they lie on the same layer and long otherwise. Thus, a proper h-layer drawing contains only proper edges, a short h-layer drawing contains no long edges, an upright h-layer drawing contains no flat edges, and an unconstrained h-layer drawing contains any type of edge. We prove

In this paper, we introduce a new graph model known as clustered graphs, i.e. graphs with recursive clustering structures. This graph model has many applications in informational and mathematical sciences. In particular, we study C-planarity of clustered graphs. Given a clustered graph, the C-planarity testing problem is to determine whether the clustered graph can be drawn without edge crossings, or edge-region crossings. In this paper, we present efficient algorithms for testing C-planarity and finding C-planar embeddings of clustered graphs. 1 Introduction Representing information visually, or by drawing graphs can greatly improve the effectiveness of user interfaces in many relational information systems [12, 17, 18, 5]. Developing algorithms for drawing graphs automatically and efficiently has become the interest of research for many computer scientists. Research in this area has been very active for the last decade. A recent survey citelabel13new of literature in this area inclu...

The barycenter heuristic is often used in practice to solve the NP-hard two-layer edge crossing minimization problem. It is well-known that the barycenter heuristic can give solutions as bad as Ω(√n) times the optimum, where n is the number of nodes in the graph. However, the example used in the proof has many isolated nodes. Makinen [8] conjectured that a better ratio bound is possible if isolated nodes are not present. We show that the ratio bound for the barycenter heuristic is still Ω(√n) even for connected bipartite graphs. We also prove a tight constant ratio bound for the barycenter heuristic on bounded-degree graphs. The bound is d - 1, where d is the maximum degree of a node in the layer that can be permuted.

Abstract. We present computational results of an implementation based on the fixed parameter tractability (FPT) approach for biplanarizing graphs. These results show that the implementation can efficiently minimum biplanarizing sets containing up to about 18 edges, thus making it comparable to previous integer linear programming approaches. We show how our implementation slightly improves the theoretical running time to O(6 bpr (G) + |G|). Finally, we explain how our experimental work predicts how performance on sparse graphs may be improved. 1

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Directed graphs are ubiquitous in most aspects of software analysis. Presented abstractly, as a list of edges, a graph does not manifest much of the important structural information that becomes obvious if the graph displayed pictorially. This paper presents a technique for drawing directed graphs quickly and attractively. It also describes how a tool implementing this technique has been used, in conjunction with other programming and analysis tools, in various aspects of software engineering. 1

A level graph G = (V; E; ) is a directed acyclic graph with a mapping : V ! f1; 2; : : : ; kg, k 1, that partitions the vertex set V as V = V 1 [V 2 [: : :[V k , V j = 1 (j), V i \V j = ; for i 6= j, such that (v) = (u)+1 for each edge (u; v) 2 E. The graph G is level planar if it can be drawn in the plane such that for each level V i , all v 2 V i are drawn on the line l i = f(x; k i) j x 2 Rg, the edges are drawn monotonically with respect to the vertical direction, and no edges intersect except at their end vertices. In this paper we give a characterization of level planar graphs in terms of minimal forbidden subgraphs called minimal level non-planar subgraph patterns (MLNP). We show that a MLNP is completely characterized by either a tree, a level non-planar cycle or a level planar cycle with certain path augmentations. These characterizations are an important first step towards attacking the NP-hard level planarization problem.

Inference of genetic networks from expression profile data is still one of the challenging works in the field of the microarray informatics. A good visualization tool would provide us a great insight into the interactions among the genes in the inferred networks. Here, we focus on the time series data of expression profiles, and discuss the method to draw a two-layered graph representing the causality