Roof Skeletons and Graph Theory Trees

The problem of constructing roofs of many papers [1; 6; 7; 9; etc.]. In this case, some studies suggested to use a computer and special programs [10; 13; 16; etc.]. The geometry of the efficient design of roofs is actual scientific direction, so in the educational process of architectural and building trades made corresponding innovations [2; 8; 14; 15; 17–20; etc.]. The roofs considered in this article are defined as special geometric polyhedral surfaces, on the basis of two assumptions: (1) all eaves of a roof form a planar (simply-connected or k-connected) polygon called the base, (2) every hipped roof end makes the same slope angle with the (horizontal) plane which contains the base. All the vertices and edges of such a roof, forming the roof skeleton, determine a graph. The orthogonal projection of a roof skeleton onto the base plane leads to a planar graph. On the basis of [11] and [12] and continuing the investigations for regular roofs, we formulate further properties which enable studying the shapes of the roofs spread over simply connected v-gons for an arbitrary integer v (v ≥ 3). We also introduce new operations: splitting and grafting of graphs, and formulate some properties of these operations. By means of such operations, the creation of a graph with cycles using two or more trees is possible. In particular, the operation of the grafting of a roof allows modeling an atrial roof.