1. Path Elongation and r- Reduced Cutting Numbers of Cycles Brad Bailey Dianna Spence North Georgia College & State University Joint Mathematics Meetings 2010

2. Agenda Background New Terms -What is a “reduced cutting number”? Min/Max Problems -Minimum or maximum edges for given cutting # Path Elongation -Initial results

3. Notation and Assumptions All graphs considered are simple and connected unless otherwise stated. V(G) = vertex set of G E(G) = edge set of G dist(u, v, G) is length of shortest path from u to v in graph G k(G) is number of components in G

4. Background Imagine: Find a parade route through a city Starts and ends at same place Does not “disconnect” city when closed to traffic Edges = Streets Vertices = Intersections

5. Definitions For a cycle C contained within a simple connected graph G, the cutting number of cycle C, denoted C#(C,G), is the number of components in G – E(C). For a simple connected graph G, the cutting number of graph G is C#(G) = max{C#(C,G) for all cycles C in G} If G is acyclic, C#(G) = 0.

6. Example  Cycle with cutting number 1  Cycle with cutting number 2  Cycle with cutting number 3  Therefore, graph has cutting number 3

7. More Definitions k r (G) denotes the number of components of G with order at least r. The graph G(C,r) is the graph that results from graph G by removing the edges of C and then deleting any components of order less than r. GG(C,3) C

8. Extension of Definition For a cycle C contained within a simple connected graph G, the r-reduced cutting number of cycle C, denoted C# r (C,G), is the number of components in G – E(C) with order at least r. G C C# 2 (C,G) = 3 C# 3 (C,G) = 1 C# 4 (C,G) = 1 C# 5 (C,G) = 0

9. Notes on r-Reduced Cutting # C# r (C,G) = k r (G – E(C)) = k(G(C,r)) The cutting number as originally defined is simply the 1-reduced cutting number.

10. Min/Max Problems Definitions m r (k,n) is the minimum number of edges in a simple connected graph on n vertices with r-reduced cutting number k M r (k,n) is the maximum number of edges in a simple connected graph on n vertices with r-reduced cutting number k

11. Results for m 1 (k,n) – Minimum m 1 (2,n) = n+2 for n  4 m 1 (k,n) = n for 3 ≤ k ≤ n...

12. Results for m r (k,n) – Minimum For, n  4, m r (2,n) = n For and, m r (k,n) = n for 3 ≤ k ≤ n

13. M r (k,n) – Maximum For n  5, M 1 (2,n) = For n  5, M r (k,n) =

14. Path Elongation Motivation: What is the impact on the length of a shortest path between two vertices when a cycle is removed? Definition: Choose vertices u and v in graph G. Then for some cycle C: I f u and v are in the same component of G-E(C), then pe(u, v, C,G) = dist(u,v,G - E(C)) – dist(u,v,G). If u and v are not in the same component of G-E(C) then we consider the path elongation to be infinite.

15. Example u v Cycle C 1 pe(u,v,C 1,G) = 5 – 3 = 2 Cycle C 2 pe(u,v,C 2,G) = infinite

16. Ready observations For complete graphs, path elongation is always less than or equal to 1. The only graphs for which path elongation is finite for all cycles and pairs of vertices are the graphs with cutting number 1.

17. Path Elongation for a pair of vertices pe 1 (u, v, G) is the maximum of pe(u, v, C, G) over all cycles C. The pe 1 (u,v,C 1,G) = 0 The pe 1 (u,v,C 2,G) = 2 u v Cycle C 1 Cycle C 2 Therefore, pe 1 (u,v,G) = 2

18. Path Elongation of a cycle pe 2 (C, G) is the maximum of pe(u,v,C,G) over all pairs of vertices u and v in G. Theorem: pe 2 (C, G) =max{pe(u,v,C,G): u and v in C}. u' v' v u

19. Path Elongations of a graph pe 1 (G) = max{pe 1 (u, v, G) : u and v in G} pe 2 (G) = max{pe 2 (C, G) : C in G} Theorem: For a connected graph G, pe 1 (G) = pe 2 (G).

20. Questions