Computing the partition function for graph homomorphisms

We introduce the partition function of edge-colored graph homomorphisms, of which the usual partition function of graph homomorphisms is a specialization, and present an efficient algorithm to approximate it in a certain domain. Corollaries include efficient algorithms for computing weighted sums approximating the number of k-colorings and the number of independent sets in a graph, as well as an efficient procedure to distinguish pairs of edge-colored graphs with many color-preserving homomorphisms G --> H from pairs of graphs that need to be substantially modified to acquire a color-preserving homomorphism G --> H.Comment: constants are improved, following a suggestion by B. Buk

Gabor Filter and Rough Clustering Based Edge Detection

This paper introduces an efficient edge detection method based on Gabor filter and rough clustering. The input image is smoothed by Gabor function, and the concept of rough clustering is used to focus on edge detection with soft computational approach. Hysteresis thresholding is used to get the actual output, i.e. edges of the input image. To show the effectiveness, the proposed technique is compared with some other edge detection methods.Comment: Proc. IEEE Conf. #30853, International Conference on Human Computer Interactions (ICHCI'13), Chennai, India, 23-24 Aug., 201

Edge Routing with Ordered Bundles

Edge bundling reduces the visual clutter in a drawing of a graph by uniting the edges into bundles. We propose a method of edge bundling drawing each edge of a bundle separately as in metro-maps and call our method ordered bundles. To produce aesthetically looking edge routes it minimizes a cost function on the edges. The cost function depends on the ink, required to draw the edges, the edge lengths, widths and separations. The cost also penalizes for too many edges passing through narrow channels by using the constrained Delaunay triangulation. The method avoids unnecessary edge-node and edge-edge crossings. To draw edges with the minimal number of crossings and separately within the same bundle we develop an efficient algorithm solving a variant of the metro-line crossing minimization problem. In general, the method creates clear and smooth edge routes giving an overview of the global graph structure, while still drawing each edge separately and thus enabling local analysis