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

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History of the n-Queens Problem 1848 – Max Bezzel 1848 – Max Bezzel 8-Queens Problem: Can eight queens be placed on an 8x8 board such that no two queens attack one another? 8-Queens Problem: Can eight queens be placed on an 8x8 board such that no two queens attack one another? 1850 – Franz Nauck 1850 – Franz Nauck n-Queens Problem: Can n queens be placed on an nxn board such that no two queens attack one another? n-Queens Problem: Can n queens be placed on an nxn board such that no two queens attack one another? 2004 – Chess Variant Pages 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? 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?

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Definition of the Queens Graph The nxn queens graph Q nxn 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) The nxn queens graph Q nxn 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) Q nxn can be broken down into rows, columns, and diagonals. Q nxn can be broken down into rows, columns, and diagonals. A complete graph K n is a graph on n vertices such that all possible edges between two vertices exist in the graph. (Blankenship 2003) A complete graph K n 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 K 4 Graphs Figure 1: Different representations of a K 4

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Number of Edges in Q nxn A complete graph on n vertices has total edges. A complete graph on n vertices has total edges. Q nxn can be broken down into rows, columns, and diagonals to determine the total number of edges. Q nxn can be broken down into rows, columns, and diagonals to determine the total number of edges. Rows:|E| = Rows:|E| = Columns:|E| = Columns:|E| = Diagonals:|E| = n(n-1) + 4 Diagonals:|E| = n(n-1) + 4 Summing the above values yields: Summing the above values yields: |E(Q nxn )| = n(n 2 -1) + 4 |E(Q nxn )| = n(n 2 -1) + 4

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Broken Down Edges of Q 4x4 Figure 2: Q 4x4 rows Figure 3: Q 4x4 columns Figure 4: Q 4x4 diagonals

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Total Edges of Q 4x4 Figure 5: Q 4x4

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Book Embeddings A book consists of a set of pages (half-planes) whose boundaries are identified on a spine (line). (Blankenship 2003) 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: 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. Each edge is assigned to exactly one page. No two edges cross in a page. No two edges cross in a page.

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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) 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. 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: Useful book thickness results: BT(G) = 1 if and only if G is outerplanar. (Gripshover 2007) BT(G) = 1 if and only if G is outerplanar. (Gripshover 2007) BT(K n ) =. (Chung, Leighton, Rosenburg 1987),(Blankenship 2003) BT(K n ) =. (Chung, Leighton, Rosenburg 1987),(Blankenship 2003)

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Book Embedding Examples Figure 6: Embedding of K 4 in, or 2 pages. (Chung, Leighton, Rosenburg 1987) Figure 7: Embedding of O 16 in one page. (Gripshover 2007)

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

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Star and Weave Patterns Figure 8: Star pattern for K 5 Figure 9: Weave pattern for Q 4x4 Figure 8: Star pattern for K 5 Figure 9: Weave pattern for Q 4x4

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Current Work: Queens Graph Upper Bound A subgraph H of a graph G has two properties: 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 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. 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) 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) Q nxn is a subgraph of the complete graph K. Q nxn is a subgraph of the complete graph K. BT(Q nxn ) ≤ BT(K ), which is equivalent to BT(Q nxn ) ≤. BT(Q nxn ) ≤ BT(K ), which is equivalent to BT(Q nxn ) ≤.

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Q 4x4 Upper Bound Figure 10: Book embedding of Q 4x4 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 graph’s outerplanarity. (Ku, Wang 2002) A maximal outerplanar graph is an outerplanar graph such that no edges can be added without violating the graph’s outerplanarity. (Ku, Wang 2002) Figure 11: Outerplanar Figure 12: Maximal outerplanar

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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. 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 Figure 13: n=8, eight adjacent vertices Figure 14: n=8, five non-adjacent vertices

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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)|. BT(G) = 1 if and only if G is outerplanar, so maximum number of edges embeddable in a single page is |E(O)|. |E(O max )| = 2n 2 -3 when |V(O max )| = n 2. |E(O max )| = 2n 2 -3 when |V(O max )| = n 2. Gripshover’s lower bound: Gripshover’s lower bound: Assumed 2n 2 -3 edges in every page Assumed 2n 2 -3 edges in every page

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

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Q 4x4 Bound Comparison Old techniques: Old techniques: 3 ≤ BT(Q 4x4 ) ≤ 13 3 ≤ BT(Q 4x4 ) ≤ 13 New techniques: New techniques: 5 ≤ BT(Q 4x4 ) ≤ 8 5 ≤ BT(Q 4x4 ) ≤ 8

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

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Single Pawn Edge Removal Conjecture: The number of edges removed depends on the dimensions of the board, the row number, of the board, the row number, and the column number: (2r+c)n - 3 - (2i-2) - (2k-3), (2r+c)n - 3 - (2i-2) - (2k-3), which is equal to (2r+c)n - 3 - c(c-1) - (r-1) 2 (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

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

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Single Pawn Upper Bound Upper bound established using complete graphs Upper bound established using complete graphs Adding a pawn similar (though not equivalent) to removing v p Adding a pawn similar (though not equivalent) to removing v p Q nxn ( p rc ) is a subgraph of K Q nxn ( p rc ) is a subgraph of K BT(Q nxn (p rc )) ≤ BT(Q nxn (p rc )) ≤ Figure 19: Edges remaining after pawn placement Figure 20: Edges remaining after removing vertex

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Summary The nxn Queens Graph Q nxn : The nxn Queens Graph Q nxn : ≤ BT(Q nxn ) ≤ ≤ BT(Q nxn ) ≤ The nxn Queens Graph After Single Pawn Placement Q nxn ( p rc ): The nxn Queens Graph After Single Pawn Placement Q nxn ( p rc ): ≤ BT(Q nxn (p rc )) ≤ ≤ BT(Q nxn (p rc )) ≤

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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, 4 th 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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