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

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?

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)

Examples of K 4 Graphs Figure 1: Different representations of a K 4

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

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

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

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.

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)

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)

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)

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

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 ) ≤.

Q 4x4 Upper Bound Figure 10: Book embedding of Q 4x4 in eight pages. (Chung, Leighton, Rosenburg 1987)

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

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

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

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 ≥.

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

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

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 (2i-2) - (2k-3), (2r+c)n (2i-2) - (2k-3), which is equal to (2r+c)n c(c-1) - (r-1) 2 (2r+c)n 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

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 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 ≥

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

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 )) ≤

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), Kelly Gripshover, The Book of Queens, preprint, Morehead State University, 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, 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),