AbstractFor a graph G, two dominating sets D and $$D'$$
D
′
in G, and a non-negative integer k, the set D is said to k-transform to $$D'$$
D
′
if there is a sequence $$D_0,\ldots ,D_\ell $$
D
0
,
…
,
D
ℓ
of dominating sets in G such that $$D=D_0$$
D
=
D
0
, $$D'=D_\ell $$
D
′
=
D
ℓ
, $$|D_i|\le k$$
|
D
i
|
≤
k
for every $$i\in \{ 0,1,\ldots ,\ell \}$$
i
∈
{
0
,
1
,
…
,
ℓ
}
, and $$D_i$$
D
i
arises from $$D_{i-1}$$
D
i
-
1
by adding or removing one vertex for every $$i\in \{ 1,\ldots ,\ell \}$$
i
∈
{
1
,
…
,
ℓ
}
. We prove that there is some positive constant c and there are toroidal graphs G of arbitrarily large order n, and two minimum dominating sets D and $$D'$$
D
′
in G such that Dk-transforms to $$D'$$
D
′
only if $$k\ge \max \{ |D|,|D'|\}+c\sqrt{n}$$
k
≥
max
{
|
D
|
,
|
D
′
|
}
+
c
n
. Conversely, for every hereditary class $$\mathcal{G}$$
G
that has balanced separators of order $$n\mapsto n^\alpha $$
n
↦
n
α
for some $$\alpha <1$$
α
<
1
, we prove that there is some positive constant C such that, if G is a graph in $$\mathcal{G}$$
G
of order n, and D and $$D'$$
D
′
are two dominating sets in G, then Dk-transforms to $$D'$$
D
′
for $$k=\max \{ |D|,|D'|\}+\lfloor Cn^\alpha \rfloor $$
k
=
max
{
|
D
|
,
|
D
′
|
}
+
⌊
C
n
α
⌋
.
On the number of minimal dominating sets on some graph classes
