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Symmetry is one of the most important aesthetic criteria on graph drawing. It is quite necessary to measure the extent to which the drawings can be considered symmetric. For this purpose, a symmetric metric based on vertex coordinate calculation is proposed in this paper. It is proven theoretically and experimentally that the proposed metric is robust to contraction, expansion, and rotation of drawings. This robustness conforms to human perception of symmetry. Star-subgraphs and cycles are two common structures in digraphs. Both of them have inherent symmetry which should be displayed in drawings. For this purpose, a force-directed algorithm named FDS is proposed which can draw star-subgraphs and cycles as symmetrically as possible. FDS algorithm draws cycles as circles whose positions are fixed to provide a scaffolding for overall layout, renders non-leaf vertices by a standard force-directed layout, and places leaf vertices on concentric circles via a deterministic strategy. A series of experiments are carried out to test FDS algorithm. The results show that FDS algorithm draws digraphs more symmetrically than the existing state-of-the-art algorithms and performs efficiency comparable to O(nlog⁡n) YFHu algorithm. [ABSTRACT FROM AUTHOR]

Rough sets theory is not good at discovering knowledge from digraphs which is a kind of relational data. In order to solve this problem, we introduce binary relations derived from simple digraphs and propose a new concept of k -step R -related set in the framework of generalized rough sets theory. In addition, we first investigate the relationships between generalized rough sets theory and graph theory on the basis of mutual representation between binary relations and digraphs. The relationships established in this work make it possible to use generalized rough sets theory to find strongly connected components of simple directed graphs, which previously can be solved only by graph algorithms. An algorithm is correspondingly developed based on the above works, especially k -step R -related set. A series of experiments are carried out to test the proposed algorithm. The results show that our algorithm provides comparable performance to the classical Tarjan algorithm. In addition, the proposed algorithm can be implemented in parallel. And its parallel performance is comparable to existing state-of-the-art parallel algorithms. [ABSTRACT FROM AUTHOR]