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The use of an alphabet of line segments to compose a curve is a possible approach for curve data compression. An existing state-of-the-art method considers a quadtree decomposition of the curve to perform the substitution of the curve parts from the alphabet of line segments. In this paper, we propose a graph theory based algorithm for tracing the curve directly to eliminate the quadtree decomposition needs. This approach obviously improves the compression efficiency, as longer line segments can be used. We tune our method further by selecting optimal turns at junctions during tracing the curve. We also discuss briefly how other application fields can take advantage of the presented approach.