If G and H are graphs, then
G
∨
H
G \vee H
is defined to be a graph obtained by identifying some edge of G with some edge of H. It is shown that for all m, n, p, and q the genus
g
(
K
m
,
n
∨
K
p
,
q
)
g({K_{m,n}} \vee {K_{p,q}})
is either
g
(
K
m
,
n
)
+
g
(
K
p
,
q
)
g({K_{m,n}}) + g({K_{p,q}})
or else
g
(
K
m
,
n
)
+
g
(
K
p
,
q
)
−
1
g({K_{m,n}}) + g({K_{p,q}}) - 1
. The latter value is attained if and only if both
K
m
,
n
{K_{m,n}}
and
K
p
,
q
{K_{p,q}}
are critical in the sense that the deletion of any edge results in a graph whose genus is one less than the genus of the original graph.