Abstract
Suppose that a biconnected graph is given, consisting of a large component plus several other smaller components, each separated from the main component by a separation pair. We investigate the existence and the computation time of schematic representations of the structure of such a graph where the main component is drawn as a disk, the vertices that take part in separation pairs are points on the boundary of the disk, and the small components are placed outside the disk and are represented as non-intersecting lunes connecting their separation pairs. We consider several drawing conventions for such schematic representations, according to different ways to account for the size of the small components. We map the problem of testing for the existence of such representations to the one of testing for the existence of suitably constrained $1$-page book-embeddings and propose several polynomial-time algorithms.
Highlights
Many of today’s applications are based on large-scale networks, having billions of vertices and edges
Examples of the second type are graph thumbnails [16], where each connected component of a graph is represented by a disk and biconnected components are represented by disks contained into the disk of the connected component they belong to
We propose to represent the large component as a disk, to draw the vertices of such a component that take part in separation pairs as points on the boundary of the disk, and to represent the small components as non-intersecting lunes connecting their separation pairs placed outside the disk
Summary
Many of today’s applications are based on large-scale networks, having billions of vertices and edges. Let us mention that our algorithms for constructing one-dimensional representations of weighted graphs only require to perform comparisons and additions between pairs of input weights; the number of bits needed to perform one of such operations can be upper bounded by a constant factor times the number of bits needed to represent two input weights, employing the real-RAM model for analyzing these algorithms might even be considered as an overkill
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