Half domination arrangements in regular and
semi-regular tessellation type graphs
Eugen J. Ionascu
Columbus State University
April 6th, 2012
http://ejionascu.ro/
Abstract:
We describe how can one use LPSolve and Maple
to provide best arrangements for the problem of
half-domination sets of vertices in transitive infinite
graphs generated by regular or semi-regular tessellations of the plane. In some cases, the results obtained are sharp. Some good upper bounds for the
average densities of vertices in half-domination sets
can be obtained by solving an appropriate optimization problem.
There are only 8 semi-regular and 4 regular tessellation
The Half-domination Problem
Best Arrangement in (36)
9
Best density 16
= 0.5625
T his is a Conjecture!
Minimal cluster tiles
(44 )
(4,82 )
,
(63 )
,
,
(36 )
,
(4,6,12)
,
(3,122 )
(33 ,42 )
,
,
(32 ,62 )
(32 ,4,3,4)
,
,
(3,4,6,4)
,
(34 ,6)
The Setting of Variables
x(0,1,1)
x(0,1,3)
x(0,0,3)
x(0,0,2)
x(0,0,4)
x(0,0,5)
x(0,0,1)
x(0,0,6)
x(0,0,1)
x(1,0,1)
x(0,0,2)
x(1,0,2)
x(1,0,1)
x(0,0,3)
x(1,0,3)
Linear programming equations (63)
3xi,j +
X
xk,l ≤ 6 for all (i,j) with 1 < i, j < n.
(k,l)∈N (i,j)
Boundary constrains for Toroidal type graphs
Case 36


2xi,j,1 + xi,j,2 + xu,j,2 + xi,v,2 ≤ 3, where u ≡ i − 1 (mod n),










 v ≡ j − 1 (mod n), u, v ∈ {0, 1, 2, ..., n − 1}



2xi,j,2 + xi,j,1 + xu,j,1 + xi,v,1 ≤ 3, where w ≡ i + 1 (mod n),








 t ≡ j + 1 (mod n), w, t ∈ {0, 1, 2, ..., n − 1}, i, j ∈ {0, 1, 2, ..., n − 1}.
(1)
Maple input for the Klein model and graph (36)
geneqtrtoridal:=proc(n)
local i,j,k,u,v,w,y,yy,obj,eq,eqq,eq1,eq2,eq3,eq4,big,x,z;
obj:=‘‘;
for i from 0 to n-1 do
for j from 0 to n-1 do
obj:=cat(obj,cat(‘x‘,i,‘y‘,j,‘z‘,1),‘+‘);
obj:=cat(obj,cat(‘x‘,i,‘y‘,j,‘z‘,2),‘+‘);
od;
od;
print(obj);
big:=‘‘;
for i from 0 to n-1 do
for j from 0 to n-1 do
u:=i-1 mod n;v:=i+1 mod n;
w:=j-1 mod n;yy:=j+1 mod n;
eq:=cat(cat(2,‘x‘,i,‘y‘,j,‘z‘,1),‘+‘,cat(‘x‘,i,‘y‘,j,‘z‘,2),‘+‘,cat(‘x‘,u,‘y‘,j,‘z‘,2),‘+‘,cat(‘x‘,i,‘y‘,w,’z’,2),‘<‘,‘3‘);
eqq:=cat(cat(2,‘x‘,i,‘y‘,j,‘z‘,2),‘+‘,cat(‘x‘,i,‘y‘,j,‘z‘,1),‘+‘,cat(‘x‘,v,‘y‘,j,‘z‘,1),‘+‘,cat(‘x‘,i,‘y‘,yy,’z’,1),‘<‘,‘3‘);
big:=cat(big,‘eq‘,‘;‘,‘eqq‘,‘;‘);
od;
od;
for i from 0 to n-1 do
for j from 0 to n-1 do
eq1:=cat(‘x‘,i,‘y‘,j,‘z‘,1,‘<‘,1);eq2:=cat(‘x‘,i,‘y‘,j,‘z‘,2,‘<‘,1);
big:=cat(big,‘eq1‘,‘;‘,‘eq2‘,‘;‘);
od;
od;
print(big);
eq4:=‘‘;
for j from 0 to n-1 do
for i from 0 to n-1 do
eq4:=cat(eq4,‘;‘,‘x‘,i,‘y‘,j,’z’,1,‘=‘,‘x‘,n-1-i mod n,‘y‘,n-1-j mod n,’z’,2);
od;
od;
print(eq4);
eq3:=‘‘;
for i from 0 to n-1 do
for j from 0 to n-1 do
eq3:=cat(eq3,‘,‘,‘x‘,i,‘y‘,j,’z’,1,‘,‘,‘x‘,i,‘y‘,j,’z’,2);
od;od;print(eq3);end:
LpSolveIDE for the Klein model and graph (36)
n
1 2 3
ρ36,n 0
1
2
5
9
4
9
16
5 6 7
14
25
5
9
27
49
8
9
9
≥ 59
16
Best arrangement (33, 42) (Conjecture)
11
Density 18
Theorem: The half-domination density for the tessellation T = (33, 42) satisfies
13 11
1
ρ(33,42) ≤
=
+
.
21 18 126
2x + x∗ ≤ 4,
2S +
X
x∗v
2xM + x∗M ≤ 3
≤ 4mn ⇒ 2S + (2S + T ) ≤ 4mn or 4S + T ≤ 4mn.
v for a squares
2T +
X
x∗v
= 6mn ⇒ 2T + (2S + 2T ) ≤ 6mn or 2S + 4T ≤ 6mn.
v for a trianlge



4x + y ≤
4
3


2x + 4y ≤ 2.
⇒



x =
5
21


y =
8
21 .
An arrangement (3, 6, 3, 6)
Density
2
3
Arrangement so far (6, 4, 3, 4)
Density
7
12
Theorem The half-domination density for the tessellation T = (3, 4, 6, 4)
satisfies
19
7
1
ρ(3,4,6,4) ≤
=
+ .
30 12 20
Best arrangements for (82, 4) and (12, 6, 4)
(a) ρ(82 ,4) = 34
(b) ρ(12,6,4) = 56
Best arrangements for (122, 3) and (4, 3, 3, 4, 3)
(a) ρ(122 ,3) = 43
(b) ρ(4,3,3,4,3) = 32
Conjectures and questions
1.
ρ1/2(G36 ) =
9
16
2.
ρ1/2(G3,6,3,6) =
3.
ρ1/2(G33,42 ) =
2
3
11
18
4. All densities are rational numbers and there exist toroidal arrangements
which attain those densities
5. Is there an approach to these problems similar to the analysis treatment of
harmonic functions?