
LinearTime Succinct Encodings of Planar Graphs via Canonical Orderings
Let G be an embedded planar undirected graph that has n vertices, m edge...
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EdgeMinimum Saturated kPlanar Drawings
For a class 𝒟 of drawings of loopless multigraphs in the plane, a drawin...
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Planar LDrawings of Bimodal Graphs
In a planar Ldrawing of a directed graph (digraph) each edge e is repre...
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Maxflow vitality in undirected unweighted planar graphs
We show a fast algorithm for determining the set of relevant edges in a ...
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Light edges in 1planar graphs of minimum degree 3
A graph is 1planar if it can be drawn in the plane so that each edge is...
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Efficient and Effective Community Search on Largescale Bipartite Graphs
Bipartite graphs are widely used to model relationships between two type...
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Implementing the Topological Model Succinctly
We show that the topological model, a semantically rich standard to repr...
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A Fast General Methodology for InformationTheoretically Optimal Encodings of Graphs
We propose a fast methodology for encoding graphs with informationtheoretically minimum numbers of bits. Specifically, a graph with property pi is called a pigraph. If pi satisfies certain properties, then an nnode medge pigraph G can be encoded by a binary string X such that (1) G and X can be obtained from each other in O(n log n) time, and (2) X has at most beta(n)+o(beta(n)) bits for any continuous superadditive function beta(n) so that there are at most 2^beta(n)+o(beta(n)) distinct nnode pigraphs. The methodology is applicable to general classes of graphs; this paper focuses on planar graphs. Examples of such pi include all conjunctions over the following groups of properties: (1) G is a planar graph or a plane graph; (2) G is directed or undirected; (3) G is triangulated, triconnected, biconnected, merely connected, or not required to be connected; (4) the nodes of G are labeled with labels from 1, ..., ell_1 for ell_1 <= n; (5) the edges of G are labeled with labels from 1, ..., ell_2 for ell_2 <= m; and (6) each node (respectively, edge) of G has at most ell_3 = O(1) selfloops (respectively, ell_4 = O(1) multiple edges). Moreover, ell_3 and ell_4 are not required to be O(1) for the cases of pi being a plane triangulation. These examples are novel applications of small cycle separators of planar graphs and are the only nontrivial classes of graphs, other than rooted trees, with known polynomialtime informationtheoretically optimal coding schemes.
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