Abstract. We present a simple encoding of plane triangulations (aka. maximal planar graphs) by plane trees with two leaves per inner node. Our encoding is a bijection taking advantage of the minimal Schnyder tree decomposition of a plane triangulation. Coding and decoding take linear time. As a byproduct we derive: (i) a simple interpretation of the formula for the number of plane triangulations with n vertices, (ii) a linear random sampling algorithm, (iii) an explicit and simple information theory optimal encoding. 1

We consider the problem of representing graphs compactly while supporting queries e#ciently. In particular we describe a data structure for representing n-vertex unlabeled graphs that satisfy an O(n )-separator theorem, c < 1. The structure uses O(n) bits, and supports adjacency and degree queries in constant time, and neighbor listing in constant time per neighbor. This generalizes previous results for graphs with constant genus, such as planar graphs.

We propose a new linear time algorithm to represent a planar graph. Based on a specific triangulation of the graph, our coding takes on average 5.03 bits per node, and 3.37 bits per node if the graph is maximal. We derive from this representation that the number of unlabeled planar graphs with n nodes is at most 2 n+O(log n) where 5.007. The current lower bound is 2 n+(log n) for 4.71. We also show that almost all unlabeled and almost all labeled n-node planar graphs have at least 1.70n edges and at most 2.54n edges.

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In previous work we described a method for compactly representing graphs with small separators, which makes use of small separators, and presented preliminary experimental results. In this paper we extend the experimental results in several ways, including extensions for dynamic insertion and deletion of edges, a comparison of a variety of coding schemes, and an implementation of two applications using the representation.

Compressed graphs representation has become an attractive research topic because of its applications in the manipulation of huge Web graphs in main memory. By far the best current result is the technique by Boldi and Vigna, which takes advantage of several particular properties of Web graphs. In this paper we show that the same properties can be exploited with a different and elegant technique, built on Re-Pair compression, which achieves about the same space but much faster navigation of the graph. Moreover, the technique has the potential of adapting well to secondary memory. In addition, we introduce an approximate Re-Pair version that works efficiently with limited main memory.

Abstract. We introduce and study orderly spanning trees of plane graphs. This algorithmic tool generalizes canonical orderings, which exist only for triconnected plane graphs. Although not every plane graph admits an orderly spanning tree, we provide an algorithm to compute an orderly pair for any connected planar graph G, consisting of an embedded planar graph H isomorphic to G, and an orderly spanning tree of H. We also present several applications of orderly spanning trees: (1) a new constructive proof for Schnyder’s realizer theorem, (2) the first algorithm for computing an area-optimal 2-visibility drawing of a planar graph, and (3) the most compact known encoding of a planar graph with O(1)-time query support. All algorithms in this paper run in linear time.

Many applications dealing with large data structures can benefit from keeping them in compressed form. Compression has many benefits: it can allow a representation to fit in main memory rather than swapping out to disk, and it improves cache performance since it allows more data to fit into the cache. However, a data structure is only useful if it allows the application to perform fast queries (and updates) to the data.

We propose a new linear time algorithm to represent a planar graph. Based on a speci c triangulation of the graph, our coding takes on average 5:03 bits per node, and 3:37 bits per node if the graph is maximal. We derive from this representation that the number of unlabeled planar graphs with n nodes is at most 2 n+O(log n) , where 5:007.

Large graph datasets are ubiquitous in many domains, including social networking and biology. Graph summarization techniques are crucial in such domains as they can assist in uncovering useful insights about the patterns hidden in the underlying data. One important type of graph summarization is to produce small and informative summaries based on userselected node attributes and relationships, and allowing users to interactively drill-down or roll-up to navigate through summaries with different resolutions. However, two key components are missing from the previous work in this area that limit the use of this method in practice. First, the previous work only deals with categorical node attributes. Consequently, users have to manually bucketize numerical attributes based on domain knowledge, which is not always possible. Moreover, users often have to manually iterate through many resolutions of summaries to identify the most interesting ones. This paper addresses both these key issues to make the interactive graph summarization approach more useful in practice. We first present a method to automatically categorize numerical attributes values by exploiting the domain knowledge hidden inside the node attributes values and graph link structures. Furthermore, we propose an interestingness measure for graph summaries to point users to the potentially most insightful summaries. Using two real datasets, we demonstrate the effectiveness and efficiency of our techniques.