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In this paper, we investigate how to topologically and geometrically characterize the intersection relations between a movable convex polygon and a set  of possibly overlapping polygons fixed in the plane. More specifically, a subset   is called an intersection relation if there exists a placement of that intersects, and only intersects, . The objective of this paper is to design an efficient algorithm that finds a finite and discrete representation of all of the intersection relations between and . Past related research only focuses on the complexity of the free space of the configuration space between and  and how to move or place an object in this free space. However, there are many applications that require the knowledge of not only the free space, but also the intersection relations. Examples are presented to demonstrate the rich applications of the formulated problem on intersection relations.

In this paper, we consider the problem of approximating the longest path in a grid graph that possesses a square-free 2-factor. By (1) analyzing several characteristics of four types of cross cells and (2) presenting three cycle-merging operations and one extension operation, we propose an algorithm that finds in an n-vertex grid graph G, a long path of length at least 5/6 n + 2. Our approximation algorithm runs in quadratic time. In addition to its theoretical value, our work has also an interesting application in image-guided maze generation.