For k ≥ 3, a convex geometric graph is called k-locally outerplanar if no path of length k intersects itself. In [D. Boutin, Convex Geometric Graphs with No Short Self-intersecting Path, Congressus Numerantium 160 (2003) 205-214], Boutin stated the results of the degeneracy for 3-locally outerplanar graphs. Later, in [D. Boutin, Structure and Properties of Locally Outerplanar Graphs, Journal of Combinatorial Mathematics and Combinatorial Computing 60 (2007) 169-180], a structural property on k-locally outerplanar graphs was proposed. These results are based on the existence of “minimal corner pairs”. In this paper, we show that a “minimal corner pair” may not exist and give a counterexample to disprove the structural property. Furthermore, we generalize the result on the degeneracy with respect to k-locally outerplanar graphs.

Digital watermarking techniques were developed for regular raster data such as images or video, but little research addressed irregular vector data, such as the shapes of cartoons or elevation contours. Vector graphic images, such as those in SVG format, are popular on the WWW, and provide the advantage of permitting affine transformations without aliasing. The creation of cartoon images or the acquisition of GIS geometry data involves much work, so the copyright and ownership of vector data must be protected. Common components in vector graphic images are polygonal lines or polylines. This work develops a normal multi-resolution representation of a polygonal line, and embeds a copyright notice or serial number in this representation. Previous studies on polyline watermarking have the non-transparent problems, including self-intersection of line segments. The experimental results demonstrate that the proposed watermarking approach is perceptually transparent, and solves the self-intersection problem. It is also resistant to similarity transformation, traversal reordering, point insertion/deletion and random noise attacks.