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Book Embeddings of Chessboard Graphs

History of the n-Queens Problem. 1848

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Book Embeddings of Chessboard Graphs

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    1. Book Embeddings of Chessboard Graphs Casey J. Hufford Morehead State University

    2. History of the n-Queens Problem 1848 Max Bezzel 8-Queens Problem: Can eight queens be placed on an 8x8 board such that no two queens attack one another? 1850 Franz Nauck n-Queens Problem: Can n queens be placed on an nxn board such that no two queens attack one another? 2004 Chess Variant Pages Pawn Placement Problem: How many pawns are necessary to place nine queens on an 8x8 board such that no two queens can attack one another?

    3. Definition of the Queens Graph The nxn queens graph Qnxn is the diagram created by connecting the vertices of two cells on a chessboard with an edge if a queen can travel from one vertex to the other in a single turn. (Gripshover 2007) Qnxn can be broken down into rows, columns, and diagonals. A complete graph Kn is a graph on n vertices such that all possible edges between two vertices exist in the graph. (Blankenship 2003)

    4. Examples of K4 Graphs Figure 1: Different representations of a K4

    5. Number of Edges in Qnxn A complete graph on n vertices has total edges. Qnxn can be broken down into rows, columns, and diagonals to determine the total number of edges. Rows: |E| = Columns: |E| = Diagonals: |E| = n(n-1) + 4 Summing the above values yields: |E(Qnxn)| = n(n2-1) + 4

    6. Broken Down Edges of Q4x4 Figure 2: Q4x4 rows Figure 3: Q4x4 columns Figure 4: Q4x4 diagonals

    7. Total Edges of Q4x4 Figure 5: Q4x4

    8. Book Embeddings A book consists of a set of pages (half-planes) whose boundaries are identified on a spine (line). (Blankenship 2003) To embed a graph in a book linearly order the vertices in the spine and assign edges to pages such that: Each edge is assigned to exactly one page. No two edges cross in a page.

    9. Book Thickness The book thickness of a graph G, denoted BT(G), is the fewest number of pages needed to embed a graph in a book over all possible vertex orderings and edge assignments. (Blankenship 2003) An outerplanar graph can be drawn in a plane such that no two edges cross and every vertex is incident with the infinite face. Useful book thickness results: BT(G) = 1 if and only if G is outerplanar. (Gripshover 2007) BT(Kn) = . (Chung, Leighton, Rosenburg 1987),(Blankenship 2003)

    10. Book Embedding Examples Figure 6: Embedding of K4 in , or 2 pages. (Chung, Leighton, Rosenburg 1987) Figure 7: Embedding of O16 in one page. (Gripshover 2007)

    11. Past Work: Queens Graph Upper Bound MSU undergraduate Kelly Gripshover: Upper bound involved a combination of graphing techniques. Star Weave Finagled (manual manipulation) Focused mainly on the 4x4 queens graph. She found that BT(Q4x4) = 13. (Gripshover 2007)

    12. Star and Weave Patterns Figure 8: Star pattern for K5 Figure 9: Weave pattern for Q4x4

    13. Current Work: Queens Graph Upper Bound A subgraph H of a graph G has two properties: The vertex set of H is a subset of the vertex set of G The edge set of H is a subset of the edge set of G. In other words, H is obtained from G by a sequence of deleting edges and vertices of G. Note that if a vertex is deleted, the edges adjacent to the vertex must also be deleted. (Bondy, Murty 1981) Qnxn is a subgraph of the complete graph K . BT(Qnxn) = BT(K ), which is equivalent to BT(Qnxn) = .

    14. Q4x4 Upper Bound Figure 10: Book embedding of Q4x4 in eight pages. (Chung, Leighton, Rosenburg 1987)

    15. Definition of Maximal Outerplanar Graph A maximal outerplanar graph is an outerplanar graph such that no edges can be added without violating the graphs outerplanarity. (Ku, Wang 2002) Figure 11: Outerplanar Figure 12: Maximal outerplanar

    16. Number of Edges in a Maximal Outerplanar Graph The number of edges in a maximal outerplanar graph on n vertices is equal to 2n-3. Figure 13: n=8, eight adjacent vertices Figure 14: n=8, five non-adjacent vertices

    17. Past Work: Queens Graph Lower Bound BT(G) = 1 if and only if G is outerplanar, so maximum number of edges embeddable in a single page is |E(O)|. |E(Omax)| = 2n2-3 when |V(Omax)| = n2. Gripshovers lower bound: Assumed 2n2-3 edges in every page

    18. Current Work: Queens Graph Lower Bound First page has 2n2-3 edges Every page after first has n2-3 edges Compare |E(Qnxn)| to maximum number of edges embeddable in a book with B pages: n(n2-1) + 4 = n2 + B(n2-3) Thus, B = .

    19. Q4x4 Bound Comparison Old techniques: 3 = BT(Q4x4) = 13 New techniques: 5 = BT(Q4x4) = 8

    20. Single Pawn Placement What effect does placing a single pawn on the board have on the upper and lower bounds? Two sets of edges are removed: All edges with the pawn vertex vp as an endpoint. All edges crossing over vp. Figure 15: Pawn blocking queen movement

    21. Single Pawn Edge Removal Conjecture: The number of edges removed depends on the dimensions of the board, the row number, and the column number: (2r+c)n - 3 - (2i-2) - (2k-3), which is equal to (2r+c)n - 3 - c(c-1) - (r-1)2 where c represents the column number, r the row, and c = r = . Figure 18: Fundamental pawn placements (unique pawn placements after any combination of rotations and reflections) for the 3x3 to 7x7 cases

    22. Single Pawn Lower Bound The number of edges remaining in Qnxn after single pawn placement is given by: [n(n2-1) + 4 ] - [(2r+c)n - 3 - c(c-1) - (r-1)2] Once again, compare |E(Qnxn(prc))| to the number of edges in a maximal outerplanar graph on n2 vertices. Thus, B =

    23. Single Pawn Upper Bound Upper bound established using complete graphs Adding a pawn similar (though not equivalent) to removing vp Qnxn(prc) is a subgraph of K BT(Qnxn(prc)) = Figure 19: Edges remaining after pawn placement Figure 20: Edges remaining after removing vertex

    24. Summary The nxn Queens Graph Qnxn: = BT(Qnxn) = The nxn Queens Graph After Single Pawn Placement Qnxn(prc): = BT(Qnxn(prc)) =

    25. References F.R.K. Chung, F.T. Leighton, and A.L. Rosenburg, Embedding Graphs in Books: A Layout Problem with Application to VLSI Design, SIAM J. Alg. Disc. Meth. 8 (1987), 33-58. Kelly Gripshover, The Book of Queens, preprint, Morehead State University, 2007. J.A. Bondy and U.S.R. Murty, Graph Theory with Applications, 4th ed. (1981). Robin Blankenship, Book Embeddings of Graphs, dissertation, Louisiana State University Baton Rouge, 2003. Shan-Chyun Ku and Biing-Feng Wang, An Optimal Simple Parallel Algorithm for Testing Isomorphism of Maximal Outerplanar Graphs, J. of Par. and Dist. Com. (2002), 221-227.

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