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Thanks to the rapid evolvement of robotic technologies, robot mowing is emerging to liberate humans from the tedious and time-consuming landscape work. Traditionally, robot mowing is perceived as a "Coverage Path Planning" problem, with a simplification that converts non-convex obstacles into convex obstacles. Besides, the converted obstacles are commonly dilated by the robot's circumcircle for collision avoidance. However when applied to robot mowing, an obstacle in a lawn is usually non-convex, imagine a garden on the lawn, such that the mentioned obstacle processing methods would fill in some concave areas so that they are not accessible to the robot anymore and hence produce inescapable uncut areas along the lawn edge, which dulls the landscape's elegance and provokes rework. To shrink the uncut area around the lawn edge we hereby reframe the problem into a brand new problem, named the "Edge Coverage Path Planning" problem that is dedicated to path planning with the objective to cover the edge. Correspondingly, we propose two planning methods, the "big and small disk" and the "sliding chopstick" planning method to tackle the problem by leveraging image morphological processing and computational geometry skills. By validation, our proposed methods can outperform the traditional "dilation-by-circumcircle" method.