Listing All Plane Graphs

In this paper we give a simple algorithm to generate all connected rooted plane graphs with at most m edges. A “rooted” plane graph is a plane graph with one designated (directed) edge on the outer face. The algorithm uses O(m) space and generates such graphs in O(1) time per graph on average without duplications. The algorithm does not output the entire graph but the difference from the previous graph. By modifying the algorithm we can generate all connected (non-rooted) plane graphs with at most m edges in O(m) time per graph. Submitted: January 2008 Reviewed: June 2008 Revised: July 2008 Accepted: November 2008 Final: December 2008 Published: February 2009 Article type: Regular Paper Communicated by: M. S. Rahman E-mail addresses: [email protected] (Katsuhisa Yamanaka) [email protected] (Shin-ichi Nakano) 6 K. Yamanaka and S. Nakano Listing All Plane Graphs Generating all graphs with some property without duplications has many applications, including unbiased statistical analysis [9]. A lot of algorithms to solve these problems are known such as [1, 2, 8, 9, 10, 15]. See textbooks [5, 6, 7, 13, 14]. In this paper we wish to generate all connected “rooted” plane graphs, which will be defined precisely in Section 2, with at most m edges. Such graphs play an important role in many algorithms, including graph drawing algorithms such as [3, 4, 12]. Using our generation algorithm one can test an implementation of a plane graph drawing algorithm. Our algorithm constructs the complete list of rooted plane graphs with at most m edges. To solve these all-graph-generating problems some types of algorithms are known. Classical method algorithms [5, p.57] first generate all the graphs with a given property allowing duplications, but output only if the graph has not been output yet. Thus this method requires quite a huge space to store a list of graphs that have already been output. Furthermore, checking whether each graph has already been output requires a lot of time. Orderly method algorithms [5, p.57] need not store the list, since they output a graph only if it is a “canonical” representative of each isomorphism class. Reverse search method algorithms [1] also need not store the list. The idea is to implicitly define a connected graph H such that the vertices of H correspond to the graphs with the given property, and the edges of H correspond to some relation between the graphs. By traversing an implicitly defined spanning tree of H, one can find all the vertices of H, which correspond to all the graphs with the given property without duplication. The main idea of our algorithms is that for some problems (biconnected triangulations [8], and triconnected triangulations [10]) we can define a tree (not a general graph) as the graph H of the reverse search method. Thus our algorithms do not need to find a spanning tree of H, since H itself is a tree. With some other ideas we give the following two simple but efficient algorithms. Our first algorithm generates all simple connected rooted plane graphs with at most m (m > 0) edges. Simple means there is neither self loops nor multiple edges. A rooted plane graph means a plane graph with one designated “root” edge on the outer face. Its precise definition is given in the next section. For instance there are nine simple connected rooted plane graphs with at most three edges, as shown in Figure 1(a). The root edges are depicted by thick grey lines. However, there are only five simple connected (non-rooted) plane graphs with at most three edges. See Figure 1(b). The algorithm uses O(m) space and runs in O(g(m)) time, where g(m) is the number of nonisomorphic connected rooted plane graphs with at most m edges. The algorithm generates each graph in O(1) time on average without duplications. The algorithm does not output the entire graph but the difference from the previous graph. By modifying the algorithm we can generate all connected (non-rooted) plane graphs with at most m edges in O(m) time per graph. The rest of the paper is organized as follows. Section 2 gives some definitions. Section 3 shows a tree structure among connected rooted plane graphs. Section JGAA, 13(1) 5–18 (2009) 7