rReduced Cutting Numbers and Cutting Powers of Cycles

r-Reduced Cutting Numbers and Cutting Powers of Cycles, Sequences of Cycles, and Graphs Brad Bailey Dianna Spence North Georgia College & State University 41 st Southeastern International Conference on Combinatorics, Graph Theory, and Computing March 12, 2010

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

Definitions l l 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}

Example Ø Ø Cycle with cutting number 1 Cycle with cutting number 2 Cycle with cutting number 3 Therefore, graph has cutting number 3

More Definitions l l kr(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. k 2(G-E(C))=3 k 3(G-E(C))=1 C G G(C, 2) G(C, 3)

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, or kr(G – E(C)). C#(C, G) = C#1(C, G) = 3 C#2(C, G) = 3 C C#3(C, G) = 1 C#4(C, G) = 1 G C#5(C, G) = 0

Extension of Definition For a simple connected graph G, the r-reduced cutting number of graph G is C#r(G) = max{C#r(C, G) for all cycles C in G} C#(G) = C#1(G) = 4 C#2(G) = 2 C#3(G) = 2 C#4(G) = 1 C#5(G) = 1

Observation For a simple connected graph G on n vertices with r-reduced cutting number k…

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

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

Results for mr(k, n) – Minimum

Mr(k, n) – Maximum For n 5, Mr (k, n) = Outline of Proof l Construction of order n graphs achieving specified number of edges with r-reduced cutting number k l Proof that C#r(G) = k holds for such graphs l Proof that such graphs have maximum possible edges for given n, r, and k

C#r(G)=k Max Edges Construction l l Have k-1 complete subgraphs of order r Have one complete subgraph of order n-r(k-1) If 2 r(k-1) n, there is a C 2 r(k-1) that does not duplicate edges of the complete subgraphs If 2 r(k-1) > n, there is a Cn with same property Kr Kr Kn-r(k-1) Kr

Cutting Power l Suppose a graph has cutting number 1 l How many cycles to “break” the graph?

Definitions l Let C = {C 1, C 2, …, Cp} be an edge-wise disjoint sequence of cycles of the graph G l These are called progressions of length p l Then C#r(C, G) is the number of components of order at least r after the removal of all the edges of C.

Cutting Power l The cutting power (at level r) of G is the length of the shortest progression of G with r-reduced cutting number at least 2.

Cutting Power l But recall the following progression. l Therefore, at levels r = 1, 2, and 3, the cutting power of K 7 is 2.

Cutting Power l Every simple connected (non-acyclic) graph has a progression of cycles with 1 -reduced cutting number at least 2. l For every simple connected (non-acyclic) graph the cutting power (at level 1) is welldefined.

Specific Structures l Cp(Kn) = l Proof Outline: When n is divisible by 4, let a = n/2 and use the fact that if a is even, Ka, a can be decomposed into a/2 = n/4 Hamiltonian cycles. Tweak decomposition for n = 4 m+r, r=1, 2, or 3 l l

Max/Min with Power l If and the maximum number of edges in a graph on n vertices with cutting power 2 is OR

Questions