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The use of an alphabet of line segments to compose a curve is a possible approach for curve data compression. Many approaches are developed with the drawback that they can process simple curves only. Curves having more sophisticated topology with self-intersections can be handled by methods considering recursive decomposition of the canvas containing the curve. In this paper, we propose a graph theory-based algorithm for tracing the curve directly to eliminate the decomposition needs. This approach obviously improves the compression performance, as longer line segments can be used. We tune our method further by selecting optimal turns at junctions during tracing the curve. We assign a polygon approximation to the curve which consists of letters coming from an alphabet of line segments. We also discuss how other application fields can take advantage of the provided curve description scheme.