Post on 14-Jun-2020
David Jenkins Advisor: Dr. DeLaViña
Domination number (γ)
2
Total Domination number (γt)
Three Domination number (γ3)
Two Domination number (γ2)
Cockayne, E.J., Dawes, R.M., Hedetniemi,
S.T., Total domination in graphs. Networks,
10:211-219, 1980.
3
Computer program written by Dr. Ermelinda
DeLaViña
Creates conjectures for specific graph
parameters
Total domination for at most cubic connected graphs
4
Graph theory directed study
Introduced to terms, notation, and proof
techniques
First conjectures from Graffiti.pc
)(Gt
t The number of minimum degree vertices
5
6
2t 2t
)(Gt )(# voft
1 G 1)(# vof
Cockayne, E.J., Dawes, R.M., & Hedetniemi,
S.T., Total domination in graphs. Networks
10:211-219, 1980.
Total domination not bounded by 2
domination
Paths and cycles are some exceptions
Interestingly Graffiti.pc conjectured the
following:
7
nt3
2
Conjecture. Let G be a connected at most
cubic graph. Then
Simplified as
Several partial results have been found
2
)(*2)( 3 G
Gt
3 t
8
If , then
If G is at most cubic then,
33
2n
9
32
n
)()( 3 GGt
Conjecture (Jenkins) Let G be an at most
cubic connected n-vertex graph.
Then G is 3-regular if and only if
The implication
if G is 3-regular, then
Is false as seen in the graph above.
)(2
3 Gn
10
)(2
3 Gn
Proposition. Let G be an at most cubic
connected n-vertex graph.
If , then G is 3-regular
Follow-up question
For a graph that is not restricted
to at most cubic, the above graph
is simple counterexample
2)(3
nG
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Proposition. Let be a smallest 3-
dominating set for G. If there is only one
not dominated by a vertex in
then
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3D
3Dv3D
3 t
13
3-dominating set
Total dominating set
14
3-dominating set
Error set
Disproved 25 different conjectures from
Graffiti.pc
Proved several partial results for
Found a partial result in a 2010 paper of Chellali
et. Al, If G is an at most cubic tree different
than a star, then
Future work could improve ‘swapping’ technique
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3 t
23 t
16