Clustered Planarity: Clusters with Few Outgoing Edges V´ ıt Jel´ ınek ⋆ , Ondˇ rej Such´ y ⋆ , Marek Tesaˇ r, and Tom´ aˇ s Vyskoˇ cil ⋆ Department of Applied Mathematics Charles University Malostransk´ e n´am. 25, 118 00 Praha, Czech Republic {jelinek,suchy,tesulo,tiger}@kam.mff.cuni.cz Abstract. We present a linear algorithm for c-planarity testing of clus- tered graphs, in which every cluster has at most four outgoing edges. 1 Introduction Clustered planarity is one of the challenges of contemporary Graph Drawing. It arises naturally when we want to draw the graph with further constraints on embedding of the vertices. This includes for example visualizing a computer network with the computers of the same department, faculty and institution being grouped together. Another application is in designing an integrated circuit with the connectors of each components being close to each other and the logical parts of the circuit being grouped together. There are many other applications including visualizations of process interaction, social networks etc. The concept of the clustered graph—a graph equipped with a system of sub- sets of vertices (called clusters), that can be recursive— was first introduced by Feng et al. in [7]. In the same paper they also proved that clustered pla- narity (shortly c-planarity) can be tested in polynomial time for c-connected clustered graphs (where each cluster induces a connected subgraph of the un- derlying graph). This was later improved by Dahlhaus [4] to a linear time al- gorithm. The paper [7] also contains a useful characterization of the c-planar graphs: Graph is c-planar if and only if there is a set of edges (usually called a saturator) that can be added to this graph to obtain a c-connected c-planar clustered graph. Since then many algorithms for testing the c-planarity were based on searching for a saturator. These include an O(n 2 )-time algorithm for ”almost” c-connected clustered graphs by Gutwenger et al. in [9,10]. An efficient algorithm for clusters with cyclic structure on a cycle was developed in [3]. The case of disjoint clusters on an embedded graph with small faces was recently addressed in [5]. Very similar result was at the same time independently published by Jel´ ınkov´ a et al. [12]. The paper [12] also contains an O(n 3 )-time algorithm for clusters of size at most three on a rib-Eulerian graph. This is an Eulerian graph that is obtained from a constant size 3-connected graph by multiplying and then subdividing edges. ⋆ Supported by grant 201/05/H014 of the Czech Science Foundation. I.G. Tollis and M. Patrignani (Eds.): GD 2008, LNCS 5417, pp. 102–113, 2009. c  Springer-Verlag Berlin Heidelberg 2009