Abstract
Let
γ
(
G
)
denote the minimum cardinality of a dominating set for
G
. A graph
G
is said to be
k
-
γ
-critical if
γ
(
G
)
=
k
, but
γ
(
G
+
e
)
<
k
for each edge
e
∈
E
(
G
¯
)
. In this paper, we provide the structure of 4-
γ
-critical connected graphs with a cut vertex. We establish that such graphs of even order contain a perfect matching. This result partially resolves a problem posed by Sumner and Wojcicka in 1998. They asked whether every
k
-
γ
-critical graph of even order contains a perfect matching for
k
≥
4
.