If
p
≤
m
p \leq m
, n then
K
m
∨
K
p
K
n
{K_m}{ \vee _{{K_p}}}{K_n}
is the graph obtained by identify ing a copy of
K
p
{K_p}
contained in
K
m
{K_m}
with a copy of
K
p
{K_p}
contained in
K
n
{K_n}
. It is shown that for all integers
p
≤
m
p \leq m
, n the genus
g
(
K
m
∨
K
p
K
n
)
g({K_m}{ \vee _{{K_p}}}{K_n})
of
K
m
∨
K
p
K
n
{K_m}{ \vee _{{K_p}}}{K_n}
is less than or equal to
g
(
K
m
)
+
g
(
K
n
)
g({K_m}) + g({K_n})
. Combining this fact with the lower bound obtained from the Euler formula, one sees that for
2
≤
p
≤
5
,
g
(
K
m
∨
K
p
K
n
)
2 \leq p \leq 5,g({K_m}{ \vee _{{K_p}}}{K_n})
is either
g
(
K
m
)
+
g
(
K
n
)
g({K_m}) + g({K_n})
or else
g
(
K
m
)
+
g
(
K
n
)
−
1
g({K_m}) + g({K_n}) - 1
. Except in a few special cases, it is determined which of these values is actually attained.