AbstractA set S of vertices in a graph G is a dominating set if every vertex not in S is ad jacent to a vertex in S. If, in addition, S is an independent set, then S is an independent dominating set. The independent domination number i(G) of G is the minimum cardinality of an independent dominating set in G. The independent domination subdivision number $$ \hbox {sd}_{\mathrm{i}}(G)$$
sd
i
(
G
)
is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the independent domination number. We show that for every connected graph G on at least three vertices, the parameter $$ \hbox {sd}_{\mathrm{i}}(G)$$
sd
i
(
G
)
is well defined and differs significantly from the well-studied domination subdivision number $$\mathrm{sd_\gamma }(G)$$
sd
γ
(
G
)
. For example, if G is a block graph, then $$\mathrm{sd_\gamma }(G) \le 3$$
sd
γ
(
G
)
≤
3
, while $$ \hbox {sd}_{\mathrm{i}}(G)$$
sd
i
(
G
)
can be arbitrary large. Further we show that there exist connected graph G with arbitrarily large maximum degree $$\Delta (G)$$
Δ
(
G
)
such that $$ \hbox {sd}_{\mathrm{i}}(G) \ge 3 \Delta (G) - 2$$
sd
i
(
G
)
≥
3
Δ
(
G
)
-
2
, in contrast to the known result that $$\mathrm{sd_\gamma }(G) \le 2 \Delta (G) - 1$$
sd
γ
(
G
)
≤
2
Δ
(
G
)
-
1
always holds. Among other results, we present a simple characterization of trees T with $$ \hbox {sd}_{\mathrm{i}}(T) = 1$$
sd
i
(
T
)
=
1
.
Independent Domination Stable Trees and Unicyclic Graphs
