r-Reduced Cutting Numbers and Cutting Powers of Cycles, Sequences of

Cycles, and Graphs

Brad BaileyDianna Spence

North Georgia College & State University

41st Southeastern International Conference onCombinatorics, Graph Theory, and Computing

March 12, 2010

Background Imagine: Find a parade route through a city

Starts and ends at same place Does not “disconnect” city when closed to traffic

Edges = StreetsVertices = Intersections

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}

Example Cycle with cutting

number 1 Cycle with cutting

number 2 Cycle with cutting

number 3 Therefore, graph

has cutting number 3

More Definitions 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.

G G(C,2)

C

G(C,3)

k2(G-E(C))=3 k3(G-E(C))=1

Extension of DefinitionFor 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)).

G

C

C#(C,G) = C#1(C,G) = 3

C#2(C,G) = 3

C#3(C,G) = 1

C#4(C,G) = 1

C#5(C,G) = 0

Extension of DefinitionFor 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…

# ( )rnC Gr

# ( )rC G k rk n

Min/Max ProblemsDefinitions mr(k,n) is the minimum number of edges in a

simple connected graph on n vertices with r-reduced cutting number k

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 m1(k,n) – Minimum

m1(2,n) = n+2 for n 4

m1 (k,n) = n for 3 ≤ k ≤ n

...

Results for mr(k,n) – Minimum

11 4, (2 , )2 rnF o r r a n d n m n n

1 3 , ( , )rn nF o r r a n d k m k n nk r

Mr(k,n) – MaximumFor n 5,

Mr (k,n) =

Outline of Proof Construction of order n graphs achieving specified

number of edges with r-reduced cutting number k Proof that C#r(G) = k holds for such graphs Proof that such graphs have maximum possible edges for

given n, r, and k

)}1(2,min{2

)1(2

)1(

krn

krnrk

C#r(G)=k Max Edges Construction

Have k-1 complete subgraphs of order r Have one complete subgraph of order n-r(k-1) If 2r(k-1) n, there is a C2r(k-1) that does not

duplicate edges of the complete subgraphs If 2r(k-1) > n, there is a Cn with same property

( 1)( 1)

2 2min{ , 2 ( 1)}

r n r kk

n r k

Kn-r(k-1) Kr

Kr

Kr

Cutting Power Suppose a graph has cutting number 1

How many cycles to “break” the graph?

Definitions Let C = {C1,C2,…, Cp} be an edge-wise

disjoint sequence of cycles of the graph G

These are called progressions of length p

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 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 But recall the following progression. Therefore, at levels r = 1, 2, and 3, the cutting

power of K7 is 2.

Cutting Power Every simple connected (non-acyclic)

graph has a progression of cycles with 1-reduced cutting number at least 2.

For every simple connected (non-acyclic) graph the cutting power (at level 1) is well-defined.

Specific Structures Cp(Kn) =

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 = 4m+r, r=1, 2, or 3

4n

Max/Min with Power If and the maximum number of

edges in a graph on n vertices with cutting power 2 is

OR

9n knr

)1(2,min22

)1(2

)1(

krn

rk

krn

)}1(2,min{),( krnnkM r

Questions