Unit Rectangle Visibility Graphs

1. Unit Rectangle Visibility Graphs Sarah Hamilton* Saint Michael’s College, Vermont Collaborating Authors: Greta Pangborn, Joanna Ellis-Monaghan, and Alice Dean * Research supported by NASA under Training Grant NGT5-40110 to the Vermont Space Grant Consortium.

2. Overview • Quick Review of Graph Theory • Review of Visibility Graphs • Introduction to Unit Rectangle Visibility Graphs • Our Results • Acknowledgements • Further Reading

3. : Cycle on n vertices : Complete graph on n vertices Trees – no cycles and connected Caterpillar – a tree that consists of a maximum length path(its spine) together with a set of vertices each of which is adjacent to a vertex on the spine. Forest – a collection of trees Important Concepts from Graph Theory

4. Bar Visibility Graphs • A Bar Visibility layout of a graph: • Represents each node as a bar in the plane • Represents edges as unobstructed vertical visibilities between the corresponding bars. • Unit Bar Visibility Graphs (UBVGs)have the additional restriction that the bars have unit length.

5. Unit Rectangle Visibility Graphs • A Rectangle Visibility layout of a graph: • Represents each node as a rectangle in the plane • Represents edges as unobstructed horizontal or vertical visibilities between the corresponding rectangles. • Unit Rectangle Visibility Graphs (URVGs) restrict the rectangles in the layout to be unit squares • URVGs more accurately model the fixed-size components needed for chip layouts

6. ?? Why Study URVGs? • Computer chip design begins with a netlist, an abstract description of thousands of logical gates with intricate wiring connections and timing constraints among them. • VLSI chip design and layout problems (BVGs and RVGs) • Fixed area and aspect ratios of computer chips • This leads us to investigate RVGs with unit rectangles as providing a more accurate model for chip design applications. • In a chip design model, the rectangles may represent gates or other components, while the visibilities represent the wires. The edges of rectangle are also of interest because they can be easily partitioned in two sets, one containing the horizontal edges, the other containing the vertical edges of a visibility layout. This is important in chip design since each layer in a chip has wires running alternately in the x or y direction.

7. Cycles • Any cycle, Cn, on n vertices is a non-collinear unit rectangle visibility graph. • If v is a vertex of a URVG G and deg(v) ≥ 7, then v is in a cycle of G. Why is this true? • GH – subgraph corresponding to horizontal edges in G • GV – subgraph corresponding to vertical edges in G • GV has UBV layout given by bottommost sides of the squares (similar for GH leftmost) Consider a URV layout where v has deg ≥ 7, • By the pigeonhole principle either GV or GH has 4 edges adjacent to v • Contradiction to Dean and Veytsel’s result that any vertex in a UBVG with deg ≥ 4 lies on a cycle. • A URVG tree T has maximum degree ≤ 6.

8. URVG Results – Complete Graphs • K4 is the largest complete graph that is a URVG. • Why isn’t K5 a URVG? • In order to layout K3 where the lower left hand corner of the squares of have coordinates (x1, y1), (x2, y2), and (x3, y3) one of the sequences (x1, x2, x3) or (y1,y2,y3) must be non-monotonic. • When you look at five squares (assuming wlog that x1 ≤ x2 ≤ x3 ≤ x4 ≤ x5) there must be a monotonic subsequence of length 3 in the sequence (y1, y2, y3, y4, y5)

9. Caterpillars • A caterpillar is a tree that consists of a maximum length path (its spine) together with set of vertices adjacent to the spine vertices. • A subdivided caterpillar allows each edge to be replaced by a path. • A caterpillar (or a subdivision of a caterpillar) is a URVG if and only if it has a spine such that deg(v) ≤ 6 for all spine vertices. • In the picture to the right, the red vertices represent the spine.

10. UBVG Trees • If T is a tree with maximum degree 3, and T is a subdivision of a caterpillar, then T is a UBVG. [Dean and Veytsel] • A maximal depth-s UBV tree has at most vertices and is isomorphic to G, which has UBV layout LG (see Figure below).

11. URVG Trees • T is a URVG tree iff it can be decomposed into two caterpillar UBVG forests where every vertex (in the UBVG forest) has deg ≤ 3 (Conjecture-Almost proved). • Given a maximal URVG tree T (depth s ≥ 1) , TV has exactly edges where Depth-1 tree Depth-2 tree

12. Complete Bipartite Graphs Km,n • Km,n (with m ≤ n) is a URVG if and only if m ≤ 2 and n ≤ 6 or m=3 and n=4. • A weak URVG layout allows extra visibilities between rectangles where there are no edges in the original graph. • Km,n (with m ≤ n) is a weak URVG if and only if m ≤ 2 (n arbitrary) or m=3 and n ≤ 4. • URV Layouts K2,6 and K3,4 • Weak URV Layout of K2,n

13. Edge Bounds If G is a URVG with n vertices, then Why is this true? Surround a URV layout with four rectangles (as in the picture on the right). Hutchinson, Shermer, and Vince prove an upper bound for the number of edges in an RVG which gives that the number of edges in our modified graph is at most 6(n+4)-20 To get a bound for the original graph we can subtract the number of edges (6) that connect nodes in the set {N, W, S, E} and a lower bound on the number of edges connecting one of the original nodes to one of the new nodes (computed by looking at the perimeter of a rectangle containing the original nodes).

14. For all n ≥ 64, there is a URVG on n vertices with at least edges. Edge Bounds

15. Acknowledgements This work was supported by VT EPSCoR under grant NSF EPS 0236976 and by NASA under Training Grant NGT5-40110 to the Vermont Space Grant Consortium.

16. Closing Remarks • New area of research • Applicability to computer chip layout • Still much more to do! • My experience as a first time researcher

17. Further Reading • For more detailed information on the proofs and ideas presented in this talk shamilton2@smcvt.edu • For more information about Erdös and Szekeres see: P. Erdos and A. Szekeres. A combinatorial problem in geometry. Compositio Mathematica, 2:463-470, 1935.

18. More information on BVGs • Alice M. Dean, William Evans, Ellen Gethner, Joshua D. Laison, Mohammad Ali Safari, and William T. Trotter. Bar k-visibility graphs. 2005. • Mike Develin, Stephen Hartke, and David Petrie Moulton. A general notion of visibility graphs. Discrete & Computational Geometry, 29(4):511-524, 2003. • Stephen G. Hartke, Jennifer Vandenbussche, and Paul Wenger. Further results on bar k-visibility graphs. 2005. • Stephen K. Wismath. Characterizing bar line-of-sight graphs. In SCG '85: Proceedings of the first annual symposium on Computational geometry, pages 147-152, New York, NY, USA, 1985. ACM Press. More information on UBVGs Alice M. Dean and Natalia Veytsel. Unit bar-visibility graphs. Congressus Numerantium, 160:161{175, 2003.

19. More information on RVGs • Prosenjit Bose, Alice M. Dean, Joan Hutchinson, and Thomas Shermer. On rectangle visibility graphs. In Graph Drawing, volume 1190, pages 25-44, 1996. • Alice M. Dean and Joan P. Hutchinson. Rectangle-visibility representations of bipartite graphs. In GD '94: Proceedings of the DIMACS International Workshop on Graph Drawing, pages 159-166, London, UK, 1995. Springer-Verlag. • Alice M. Dean and Joan P. Hutchinson. Rectangle-visibility layouts of unions and products of trees. Journal of Graph Algorithms and Applications, 2(8):1-21, 1998. • Thomas Shermer. On rectangle visibility graphs ii: k-hilly and maximum-degree 4. 1996. • Ileana Streinu and Sue Whitesides. Rectangle visibility graphs: Characterization, construction, and compaction. In STACS '03: Proceedings of the 20th Annual Symposium on Theoretical Aspects of Computer Science, pages 26-37, London, UK, 2003. Springer-Verlag.

20. More information on RVGs in 3 dimensions • Prosenjit Bose, Hazel Everett, Sandor Fekete, Anna Lubiw, Henk Meijer, Kathleen Romanik, Tom Shermer, and Sue Whitesides. On a visibility representation for graphs in three dimensions. 1995. • Michael Burstein and Richard Pelavin. Hierarchical channel router. In DAC '83: Proceedings of the 20th conference on Design automation, pages 591-597, Piscataway, NJ, USA, 1983. IEEE Press. • Olivier Devillers, Hazel Everett, Sylvian Lazard, Maria Pentcheva, and Steven Wismath. Drawing kn in three dimensions with one bend per edge. INRIA, (5708), September 2005. • Vida Dujmovic', David Eppstein, Matthew Suderman, and David R. Wood. Drawings of planar graphs with few slopes and segments, 2006. • Sung-Chuan Fang, Kuo-En Chang, Wu-Shiung Feng, and Sao-Jie Chen. Constrained via minimization with practical considerations for multi-layer vlsi/pcb routing problems. In DAC '91: Proceedings of the 28th conference on ACM/IEEE design automation, pages 60-65, New York, NY, USA, 1991. ACM Press. • Sandor P. Fekete and Henk Meijer. Rectangle and box visibility graphs in 3d. International Journal of Computational Geometry and Applications, 9(1):1-21, 1999. • D. R. Wood. Three-Dimensional Orthogonal Graph Drawing. School of Computer Science and Software Engineering, Monash University, Australia 3168, 2000.

21. More information on Thickness of Graphs • Michael B. Dillencourt, David Eppstein, and Daniel S. Hirschberg. Geometric thickness of complete graphs. Journal of Graph Algorithms and Applications, 4(3):5-17, 2000. • Joan P. Hutchinson, Thomas Shermer, and Andrew Vince. On representations of some thickness-two graphs. Computational Geometry, 13:161-171, 1999. • Petra Mutzel, Thomas Odenthal, and Mark Scharbrodt. The thickness of graphs: A survey. 1998. • David R. Wood and Jan Arne Telle. Planar decompositions and the crossing number of graphs with an excluded minor, 2006.

22. Thank You for Coming!