Book Embeddings Book Embeddings of Chessboard of Chessboard

GraphsGraphs

Casey J. HuffordCasey J. HuffordMorehead State UniversityMorehead State University

History of the n-Queens History of the n-Queens ProblemProblem

1848 – Max Bezzel1848 – Max Bezzel 8-Queens Problem: Can eight queens be placed on an 8-Queens Problem: Can eight queens be placed on an

8x8 board such that no two queens attack one 8x8 board such that no two queens attack one another?another?

1850 – Franz Nauck1850 – Franz Nauck n-Queens Problem: Can n queens be placed on an n-Queens Problem: Can n queens be placed on an

nxn board such that no two queens attack one nxn board such that no two queens attack one another?another?

2004 – 2004 – Chess Variant PagesChess Variant Pages Pawn Placement Problem: How many pawns are Pawn Placement Problem: How many pawns are

necessary to place nine queens on an 8x8 board such necessary to place nine queens on an 8x8 board such that no two queens can attack one another?that no two queens can attack one another?

Definition of the Queens Definition of the Queens GraphGraph

The The nxn queens graph Qnxn queens graph Qnxnnxn is the diagram created is the diagram created by connecting the vertices of two cells on a by connecting the vertices of two cells on a chessboard with an edge if a queen can travel chessboard with an edge if a queen can travel from one vertex to the other in a single turn. from one vertex to the other in a single turn. (Gripshover 2007)(Gripshover 2007)

QQnxnnxn can be broken down into rows, columns, and can be broken down into rows, columns, and diagonals.diagonals.

A A complete graph Kcomplete graph Knn is a graph on n vertices such is a graph on n vertices such that all possible edges between two vertices exist that all possible edges between two vertices exist in the graph. in the graph. (Blankenship 2003)(Blankenship 2003)

Examples of KExamples of K44 Graphs Graphs

Figure 1: Different representations of a KFigure 1: Different representations of a K44

Number of Edges in QNumber of Edges in Qnxnnxn

A complete graph on n vertices has total A complete graph on n vertices has total edges.edges.

QQnxnnxn can be broken down into rows, columns, and can be broken down into rows, columns, and diagonals to determine the total number of edges.diagonals to determine the total number of edges. Rows:Rows: |E| = |E| =

Columns:Columns: |E| = |E| =

Diagonals:Diagonals: |E| = n(n-1) + 4 |E| = n(n-1) + 4

Summing the above values yields:Summing the above values yields: |E(|E(QQnxnnxn)| = n(n)| = n(n22-1) + -1) + 4 4

2)1( nn

2)1(2 nn

2)1(2 nn

1

2 2)1(n

i

ii

1

2 2)1(n

i

ii

Broken Down Edges of QBroken Down Edges of Q4x44x4

Figure 2: QFigure 2: Q4x44x4 rows Figure 3: Q rows Figure 3: Q4x44x4 columns columns

Figure 4: QFigure 4: Q4x44x4 diagonals diagonals

Total Edges of QTotal Edges of Q4x44x4

Figure 5: QFigure 5: Q4x44x4

Book EmbeddingsBook Embeddings

A A book book consists of a set of pages (half-consists of a set of pages (half-planes) whose boundaries are identified on planes) whose boundaries are identified on a spine (line). a spine (line). (Blankenship 2003)(Blankenship 2003)

To To embed embed a graph in a book linearly order a graph in a book linearly order the vertices in the spine and assign edges the vertices in the spine and assign edges to pages such that: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 ThicknessBook Thickness

The The book thickness of a graph Gbook thickness of a graph G, denoted , denoted BT(G)BT(G), is the fewest number of pages needed , is the fewest number of pages needed to embed a graph in a book over all possible to embed a graph in a book over all possible vertex orderings and edge assignments. vertex orderings and edge assignments. (Blankenship 2003)(Blankenship 2003)

An An outerplanar graph outerplanar graph can be drawn in a plane can be drawn in a plane such that no two edges cross and every such that no two edges cross and every vertex is incident with the infinite face.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. BT(G) = 1 if and only if G is outerplanar. (Gripshover 2007)(Gripshover 2007)

BT(KBT(Knn) = . ) = . (Chung, Leighton, Rosenburg 1987),(Blankenship 2003)(Chung, Leighton, Rosenburg 1987),(Blankenship 2003)

2n

Book Embedding ExamplesBook Embedding ExamplesFigure 6: Embedding of KFigure 6: Embedding of K44 in , or 2 pages. in , or 2 pages.

(Chung, Leighton, Rosenburg 1987)(Chung, Leighton, Rosenburg 1987)

Figure 7: Embedding of OFigure 7: Embedding of O1616 in one page. in one page. (Gripshover 2007)(Gripshover 2007)

24

Past Work:Past Work:Queens Graph Upper BoundQueens Graph Upper Bound

MSU undergraduate Kelly Gripshover:MSU undergraduate Kelly Gripshover: Upper bound involved a combination of Upper bound involved a combination of

graphing techniques.graphing techniques. StarStar WeaveWeave Finagled (manual manipulation)Finagled (manual manipulation)

Focused mainly on the 4x4 queens graph. She Focused mainly on the 4x4 queens graph. She found that BT(Qfound that BT(Q4x44x4) ) ≤ 13. ≤ 13. (Gripshover 2007)(Gripshover 2007)

Star and Weave PatternsStar and Weave Patterns

Figure 8: Star pattern for KFigure 8: Star pattern for K55 Figure 9: Weave pattern for Figure 9: Weave pattern for

QQ4x44x4

Current Work:Current Work:Queens Graph Upper BoundQueens Graph Upper Bound

A A subgraph H of a graph Gsubgraph H of a graph G has two properties: has two properties: The vertex set of H is a subset of the vertex set of GThe 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 In other words, H is obtained from G by a sequence of

deleting edges and vertices of G. Note that if a vertex is deleting edges and vertices of G. Note that if a vertex is deleted, the edges adjacent to the vertex must also be deleted, the edges adjacent to the vertex must also be deleted. deleted. (Bondy, Murty 1981) (Bondy, Murty 1981)

QQnxnnxn is a subgraph of the complete graph K . is a subgraph of the complete graph K . BT(QBT(Qnxnnxn) ) ≤ BT(≤ BT(K K ), which is equivalent to BT(), which is equivalent to BT(QQnxnnxn) ≤ . ) ≤ .

2n

2n

2

2n

QQ4x44x4 Upper Bound Upper Bound

Figure 10: Book embedding of QFigure 10: Book embedding of Q4x44x4 in eight pages. in eight pages. ((Chung, Leighton, Rosenburg 1987)Chung, Leighton, Rosenburg 1987)

Definition of Definition of Maximal Outerplanar GraphMaximal Outerplanar Graph

A A maximal outerplanar graph maximal outerplanar graph is an outerplanar is an outerplanar graph such that no edges can be added without graph such that no edges can be added without violating the graph’s outerplanarity. violating the graph’s outerplanarity. (Ku, Wang 2002)(Ku, Wang 2002)

Figure 11: Outerplanar Figure 12: Maximal outerplanarFigure 11: Outerplanar Figure 12: Maximal outerplanar

Number of Edges in a Number of Edges in a Maximal Outerplanar GraphMaximal Outerplanar Graph

The number of edges in a maximal The number of edges in a maximal outerplanar graph on n vertices is equal to outerplanar graph on n vertices is equal to 2n-3.2n-3.

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

Past Work:Past Work:Queens Graph Lower BoundQueens Graph Lower Bound

BT(G) = 1 if and only if G is outerplanar, so BT(G) = 1 if and only if G is outerplanar, so maximum number of edges embeddable in maximum number of edges embeddable in a single page is |E(O)|.a single page is |E(O)|.

|E(O|E(Omaxmax)| = 2n)| = 2n22-3 when |V(O-3 when |V(Omaxmax)| = n)| = n22..

Gripshover’s lower bound:Gripshover’s lower bound: Assumed 2nAssumed 2n22-3 edges in every page-3 edges in every page

Current Work:Current Work:Queens Graph Lower BoundQueens Graph Lower Bound

First page has 2nFirst page has 2n22-3 edges-3 edges

Every page after first has nEvery page after first has n22-3 edges-3 edges

Compare |E(QCompare |E(Qnxnnxn)| to maximum number of edges )| to maximum number of edges embeddable in a book with B pages:embeddable in a book with B pages:

n(nn(n22-1) + -1) + 4 4 ≤ n≤ n22 + B(n + B(n22-3)-3)

Thus, B ≥ .Thus, B ≥ .

1

2 2)1(n

i

ii

32

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2

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iinnn

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QQ4x4 4x4 Bound ComparisonBound Comparison

Old techniques:Old techniques: 3 ≤ BT(Q3 ≤ BT(Q4x44x4) ≤ 13) ≤ 13

New techniques:New techniques: 5 ≤ BT(Q5 ≤ BT(Q4x44x4) ≤ 8) ≤ 8

Single Pawn PlacementSingle Pawn Placement

What effect does placing a single pawn on the What effect does placing a single pawn on the board have on the upper and lower bounds?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 vAll edges with the pawn vertex vpp as an endpoint. as an endpoint.

All edges “crossing over” vAll edges “crossing over” vpp..

Figure 15: Pawn blocking queen movementFigure 15: Pawn blocking queen movement

Single Pawn Edge RemovalSingle Pawn Edge RemovalConjecture:Conjecture: The number of edges The number of edges

removed depends on the dimensionsremoved depends on the dimensions of the board, the row number, of the board, the row number,

and the column number: and the column number:

(2r+c)n - 3 - (2i-2) - (2k-3),(2r+c)n - 3 - (2i-2) - (2k-3),

which is equal towhich is equal to

(2r+c)n - 3 - c(c-1) - (r-1)(2r+c)n - 3 - c(c-1) - (r-1)22

where c represents the column number, where c represents the column number,

r the row, and c ≤ r ≤ .r the row, and c ≤ r ≤ .

Figure 18: Fundamental pawn placements (unique pawn placements after any Figure 18: Fundamental pawn placements (unique pawn placements after any combinationcombination of rotations and reflections) for the 3x3 to 7x7 cases of rotations and reflections) for the 3x3 to 7x7 cases

c

i 2

r

k 2

2n

Single Pawn Lower BoundSingle Pawn Lower Bound

The number of edges remaining in QThe number of edges remaining in Qnxnnxn after single after single pawn placement is given by:pawn placement is given by:

[[n(nn(n22-1) + -1) + 4 4 ] - [(2r+c)n - 3 - c(c-1) - (r-1)] - [(2r+c)n - 3 - c(c-1) - (r-1)22]]

Once again, compare |E(QOnce again, compare |E(Qnxnnxn((pprcrc))| to the number of ))| to the number of edges in a maximal outerplanar graph on nedges in a maximal outerplanar graph on n22 vertices. vertices.

Thus, B ≥ Thus, B ≥

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Single Pawn Upper BoundSingle Pawn Upper Bound Upper bound established using complete graphsUpper bound established using complete graphs Adding a pawn similar (though not equivalent) to Adding a pawn similar (though not equivalent) to

removing vremoving vpp

QQnxnnxn((pprcrc) is a subgraph of K) is a subgraph of K BT(QBT(Qnxnnxn(p(prcrc)) ≤)) ≤

Figure 19: Edges remaining after pawn placementFigure 19: Edges remaining after pawn placement

Figure 20: Edges remaining after removing vertexFigure 20: Edges remaining after removing vertex

12 n

2

12n

SummarySummary

The nxn Queens Graph QThe nxn Queens Graph Qnxnnxn:: ≤ ≤ BT(QBT(Qnxnnxn) ≤ ) ≤

The nxn Queens Graph After Single Pawn The nxn Queens Graph After Single Pawn Placement QPlacement Qnxnnxn((pprcrc):): ≤ ≤ BT(QBT(Qnxnnxn(p(prcrc)) ≤)) ≤

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ReferencesReferences

F.R.K. Chung, F.T. Leighton, and A.L. Rosenburg, F.R.K. Chung, F.T. Leighton, and A.L. Rosenburg, Embedding Embedding Graphs in Books: A Layout Problem with Application to VLSI Graphs in Books: A Layout Problem with Application to VLSI DesignDesign, SIAM J. Alg. Disc. Meth. 8 (1987), 33-58., SIAM J. Alg. Disc. Meth. 8 (1987), 33-58.

Kelly Gripshover, Kelly Gripshover, The Book of QueensThe Book of Queens, preprint, Morehead , preprint, Morehead State University, 2007.State University, 2007.

J.A. Bondy and U.S.R. Murty, J.A. Bondy and U.S.R. Murty, Graph Theory with ApplicationsGraph Theory with Applications, , 44thth ed. (1981). ed. (1981).

Robin Blankenship,Robin Blankenship, Book Embeddings of Graphs Book Embeddings of Graphs, dissertation, , dissertation, Louisiana State University – Baton Rouge, 2003.Louisiana State University – Baton Rouge, 2003.

Shan-Chyun Ku and Biing-Feng Wang, Shan-Chyun Ku and Biing-Feng Wang, An Optimal Simple An Optimal Simple Parallel Algorithm for Testing Isomorphism of Maximal Parallel Algorithm for Testing Isomorphism of Maximal Outerplanar GraphsOuterplanar Graphs, J. of Par. and Dist. Com. (2002), 221-, J. of Par. and Dist. Com. (2002), 221-227.227.