The Equivalence Number and Transit Graphs for Chessboard Graphs

1. The Equivalence Number and Transit Graphs for Chessboard Graphs B. Nicholas Wahle Morehead State University

2. Graphs A graph is a set of points called vertices with unordered pairs of vertices called edges.

3. Paths A path is a subset of the vertices such that there is an edge connecting one vertex to the next. The vertices and edges in red form a P4. Explain the pictures of the path, say notationExplain the pictures of the path, say notation

4. Complete Graph A complete graph, is a graph in which all vertices are adjacent to every other vertex by an edge. This graph forms a K6. Describe a Complete Graph, say notationDescribe a Complete Graph, say notation

5. Cliques A clique is a subset of the vertices such that the subset forms a complete graph. The vertices and edges in red form a clique of order 5. Explain what a clique isExplain what a clique is

6. Independence An independent set of vertices in a graph is a set such that none of the vertices are joined by an edge. The independence number of the graph is the largest number of independent vertices that can be found. The vertices in red form an independent set.

7. N-Queens Problem The original Queens problem asked if eight queens could be placed on a standard 8x8 chessboard such that no two queens attack each other. (Bezzel, 1848) It was later generalized as N queens being placed on a NxN chessboard for N larger than 4. (Nauck, 1850)

8. N+k Queens Problem The NxN board could not contain more than N queens, since a queen can attack any space in its row. More queens can be added to the board by placing pawns to block their attacks. Given a large enough N, it has been shown that N+k queens can be placed on an NxN board with k pawns separating them. (Chatham, et al 2006)

9. Queens Graph A queens graph is a graph where each square of a chessboard is represented by a vertex in the graph. The graph has an edge connecting two vertices if a queen can move from one square to the other in a legal move. N-Queens is an independence problem on the queens graphN-Queens is an independence problem on the queens graph

10. Pieces Paths that a queen can move along in a single turnPaths that a queen can move along in a single turn

11. Transit Graphs Let F be a family of graphs on the same vertex set, V. The transit graph of F is the graph on V such that ab is an edge if and only if there is a path from a to b in one of the graphs of F. The elements of F are called routes. The equivalence number of a graph, eq(G), is the minimum number of routes required to construct the graph G.

12. Another Look The routes are similar to a subway map or a road map. The maps show you where you can go without having to change roads or subway trains. Image from http://www.rususa.com/ Consider the transit graph of this mapConsider the transit graph of this map

13. A Minimal Example non-minimumnon-minimum

14. A Minimum Example

15. Covering a Vertex Given a vertex v, define c(v) to be the minimum number of cliques required to cover all edges incident with vertex v. Define C(G) to be the maximum c(v) of all the vertices of the graph G.

16. Finding C(G) C(G)=3 Mention it is a Max of the Min’sMention it is a Max of the Min’s

17. Bounds on Equivalence We have shown that C(G) = eq(G) and conjectured that eq(G) = C(G) + 1 Since C(G) cliques are required to cover at least one vertex and at most one clique containing that vertex can be represented in a route, C(G) = eq(G). Currently there is not a proof for eq(G) = C(G)+1 nor has it been disproved.

18. Why? Subway analogy, a station is shut down, Mention the eq(Q)=4 for n>3… This is not a MinimumSubway analogy, a station is shut down, Mention the eq(Q)=4 for n>3… This is not a Minimum

19. Remove a Vertex Subway analogy, a station is shut downSubway analogy, a station is shut down

20. Putting Together the Pieces Left is with the vertex removed, right side is the original, doted on the left is what would be left if it were purely removing a vertexLeft is with the vertex removed, right side is the original, doted on the left is what would be left if it were purely removing a vertex

21. Other Chess Pieces For the queens graph, the equivalence number is 4, for a 4x4 or larger board. The rooks graph has an equivalence number of 2, for 2x2 board or larger. For a 3x3 board or larger, the bishops graph has an equivalence number of 2. A knights graph has an equivalence number of 8, for a board 5x5 or larger.

22. Knights Graph The knights graph does not allow for a clique larger than a K2. Therefore c(v) is equal to the number of edges incident on that vertex. Largest c(v) is the center point, C(G)=8Largest c(v) is the center point, C(G)=8

23. References http://npluskqueens.info R.D. Chatham, G.H. Fricke, and R.D. Skaggs, The Queens Separation Problem, Util. Math. 69 (2006), 129-141 Chatham, Douglas, et al, Independence and Domination on Chessboard Graphs, preprint, Morehead State University, 2006 Frankl, Peter, Covering Graphs by Equivalence Relations. Annals of Discrete Mathematics 12 (1982): 125-127 Harless, Joe, Transit Graphs: Separation, Domination, and Other Parameters, preprint, Morehead State University, 2007 Joe’s Capstone, Morehead Papers(2)Joe’s Capstone, Morehead Papers(2)