Book Embeddings of Chessboard Graphs

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)

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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

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Broken Down Edges of Q4x4

• Figure 2 Q4x4 rows Figure 3 Q4x4
  columns

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Total Edges of 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)

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Book Embedding Examples

• Figure 6 Embedding of K4 in , or 2 pages.
  (Chung, Leighton, Rosenburg 1987)

11
Past WorkQueens 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)

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Star and Weave Patterns

• Figure 8 Star pattern for K5
  Figure 9 Weave pattern for Q4x4

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Current WorkQueens 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) .

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Q4x4 Upper Bound

• Figure 10 Book embedding of Q4x4 in eight pages.
  (Chung, Leighton, Rosenburg 1987)

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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)

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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 n8, eight adjacent vertices
  Figure 14 n8, five non-adjacent vertices
  

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Past WorkQueens 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

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Current WorkQueens 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
• (2rc)n - 3 - (2i-2) - (2k-3),
• which is equal to
• (2rc)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 - (2rc)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

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Summary

• The nxn Queens Graph Qnxn
• BT(Qnxn)
  
• The nxn Queens Graph After Single Pawn Placement
  Qnxn(prc)
• 
  BT(Qnxn(prc))

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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.