Hierarchical graphs and clustered graphs are useful non-classical graph models for structured relational information. Hierarchical graphs are graphs with layering structures; clustered graphs are graphs with recursive clustering structures. Both have applications in CASE tools, software visualization, and VLSI design. Drawing algorithms for hierarchical graphs have been well investigated. However, the problem of straight-line representation has not been solved completely. In this paper, we answer the question: does every planar hierarchical graph admit a planar straight-line hierarchical drawing? We present an algorithm that constructs such drawings in linear time. Also, we answer a basic question for clustered graphs, that is, does every planar clustered graph admit a planar straight-line drawing with clusters drawn as convex polygons? We provide a method for such drawings based on our algorithm for hierarchical graphs.

Abstract. A clustered graph has its vertices grouped into clusters in a hierarchical way via subset inclusion, thereby imposing a tree structure on the clustering relationship. The c-planarity problem is to determine if such a graph can be drawn in a planar way, with clusters drawn as nested regions and with each edge (drawn as a curve between vertex points) crossing the boundary of each region at most once. Unfortunately, as with the graph isomorphism problem, it is open as to whether the cplanarity problem is NP-complete or in P. In this paper, we show how to solve the c-planarity problem in polynomial time for a new class of clustered graphs, which we call extrovert clustered graphs. This class is quite natural (we argue that it captures many clustering relationships that are likely to arise in practice) and includes the clustered graphs tested in previous work by Dahlhaus, as well as Feng, Eades, and Cohen. Interestingly, this class of graphs does not include, nor is it included by, a class studied recently by Gutwenger et al.; therefore, this paper offers an alternative advancement in our understanding of the efficient drawability of clustered graphs in a planar way. Our testing algorithm runs in O(n 3) time and implies an embedding algorithm with the same time complexity. 1

Graphs are often used to represent relational information. As the amount of information that we want to visualize becomes larger and more complicated, classical graph model tends to be insufficient. In this paper, we introduce and show how to draw a practical and simple graph structure known as clustered graphs. We present an algorithm which produces planar, straight-line, convex drawings of clustered graphs in O(n 2:5 ) time. We also demonstrate an area lower bound and an angle upper bound for straight-line convex drawings of C-planar graphs. We show that such drawings require\Omega\Gammaq n ) area and the smallest angle is O(1=n). Our bounds are unlike the area and angle bounds of classical graph drawing conventions in which area bound is \Omega\Gamma n 2 ) and angle bounds are functions of the maximum degree of the graph. Our results indicate important tradeoffs between line straightness and area, and between region convexity and area. 1 Introduction Many systems, particular...

Abstract- The problem of testing c-planarity of c-graphs is unknown to be NP-complete or in P. Previous work solved this problem on some special classes of c-graphs. In particular, Goodrich, Lueker, and Sun tested c-planarity of extrovert c-graphs in O(n 3) time [5]. In this paper, we improve the time complexity of the testing algorithm in [5] to O(n) 2. Keywords: