Which selection method examines the shape of geometries based on edges and vertices?

The selected answer accurately identifies the method that specifically examines the shape of geometries through their edges and vertices. In GIS terminology, this involves determining relationships based on the physical outlines and connections of geometric features.

The phrase "share a line segment with" signifies an examination of geometries that directly interact or connect along the same linear segment—an essential aspect when analyzing how features relate to one another in a spatial context. This connection is not just about proximity but focuses on the exact points where these features are congruent along their edges.

The other options, while related to spatial relationships, imply different interactions that do not specifically focus on the geometrical edges and vertices. For instance, "contain Clementini" may pertain to a classification system and involves the concept of one geometry fully encompassing another but does not specifically address edges. "Are within" indicates an entirely contained relationship without considering the border connection, and "touch the boundary of" would be more inclusive of boundary relationships without enforcing a direct shared geometry condition. By focusing on shared line segments, the selected answer effectively encapsulates the essence of spatial analysis related to geometrical shapes and their direct interactions.