Search for: All records

Abstract Graph sampling methods have been used to reduce the size and complexity of big complex networks for graph mining and visualization. However, existing graph sampling methods often fail to preserve the connectivity and important structures of the original graph. This paper introduces a new divide and conquer approach to spectral graph sampling based on graph connectivity, called the BC Tree (i.e., decomposition of a connected graph into biconnected components) and spectral sparsification. Specifically, we present two methods, spectral vertex sampling $$BC\_SV$$ B C _ S V and spectral edge sampling $$BC\_SS$$ B C _ S S by computing effective resistance values of vertices and edges for each connected component. Furthermore, we present $$DBC\_SS$$ D B C _ S S and $$DBC\_GD$$ D B C _ G D , graph connectivity-based distributed algorithms for spectral sparsification and graph drawing respectively, aiming to further improve the runtime efficiency of spectral sparsification and graph drawing by integrating connectivity-based graph decomposition and distributed computing. Experimental results demonstrate that $$BC\_SV$$ B C _ S V and $$BC\_SS$$ B C _ S S are significantly faster than previous spectral graph sampling methods while preserving the same sampling quality. $$DBC\_SS$$ D B C _ S S and $$DBC\_GD$$ D B C _ G D obtain further significant runtime improvement over sequential approaches, and $$DBC\_GD$$ D B C _ G D further achieves significant improvements in quality metrics over sequential graph drawing layouts. 

This paper introduces and studies the following beyond-planarity problem, which we call h-Clique2Path Planarity. Let G be a simple topological graph whose vertices are partitioned into subsets of size at most h, each inducing a clique. h-Clique2Path Planarity asks whether it is possible to obtain a planar subgraph of G by removing edges from each clique so that the subgraph induced by each subset is a path. We investigate the complexity of this problem in relation to k-planarity. In particular, we prove that h-Clique2Path Planarity is NP-complete even when h=4 and G is a simple 3-plane graph, while it can be solved in linear time when G is a simple 1-plane graph, for any value of h. Our results contribute to the growing fields of hybrid planarity and of graph drawing beyond planarity.