1. Roofs and Graph-theoretical Trees Edwin KOŹNIEWSKI Białystok University of Technology Poland edwikozn@pb.bialystok.pl

2. Roofs shape the lansdcape The paper is a continuation of research concerning with geometrical properties of roofs of buildings considered as a special class of polyhedral surfaces from the view point of Graph Theory. The new concepts (dividing and grafting of trees, primitives of roofs, decomposition and joining of roofs) are introduced and appropriate properties are formulated and proved. These concepts and properties lead to a description of shapes of regular roofs spread over simply connected v-gons for arbitrary integer v (v  3)

3. Classification of solutions of shapes of a regular roof generated by a simply connected v-gon for v=3,4,5,6,7,8 Student’s works – 2002-2004. Civil Engineering Faculty, I semester Bialystok University of Technology Poland Fragment „Geometry of Roofs” of Exhibition „Mathematics for artists, mathematics for everyone”. Bialystok University of Technology 2004., Poland””.

4. The regular roof generated by a 10-gon. Student’s work – 2004 Civil Engineering Faculty, I semester. Bialystok University of Technology, Poland

5. Roofs and their skeletons

6. The part regular roof (with court-yard) of the building of the Białystok University and the block of atrial buildings of the Theological Seminary:

7. We introduce two new operations in Graph Theory: dividing and grafting of trees.

8. The least nonelementary tree and roofs The elementary trees of degree at most 3 and suitable basic (elementary) roofs

12. (A) (B) (C)

14. In a proof of this theorem we use the geometric operations: „ccc – chamfering of a convex corner ” „mcc – making a concave corner ”

17. Conformal transformations of roofs scaling

18. Conformal transformations of roofs translation

21. A1) twice used scaling A2) dividing of the roof into three roofs A3) three elementary roofs

22. A4) creation of the primitives of roofs

23. A5) primitives of elementary roofs for the roof A

24. Thank you very much