Concept:
Indian classical and traditional theatre forms are closely associated with particular dance traditions and regional performance styles. Many dramatic forms contributed to the evolution of present-day classical dances.
Step 1: Match Koodiattam.
\[
\text{Koodiattam}
\]
is the ancient Sanskrit theatre tradition of Kerala closely related to:
\[
\text{Kathakali}
\]
Thus:
\[
A \rightarrow IV
\]
Step 2: Match Yakshagana.
\[
\text{Yakshagana}
\]
is a dance-drama tradition that influenced:
\[
\text{Kuchipudi}
\]
Thus:
\[
B \rightarrow III
\]
Step 3: Match Bhagavata Mela Natakam.
\[
\text{Bhagavata Mela Natakam}
\]
is associated with devotional dance-drama traditions connected to:
\[
\text{Bharata Natyam}
\]
Thus:
\[
C \rightarrow II
\]
Step 4: Match Natavari Nritya.
\[
\text{Natavari Nritya}
\]
is a rhythmic dance style associated with:
\[
\text{Kathak}
\]
Thus:
\[
D \rightarrow I
\]
Step 5: Write the final matching sequence.
The correct matching is:
\[
A\text{-}IV,\ B\text{-}III,\ C\text{-}II,\ D\text{-}I
\]
Step 6: Compare with the given options.
Among the given choices:
\[
(A)\ A\text{-}IV,\ B\text{-}III,\ C\text{-}II,\ D\text{-}I
\]
\[
(B)\ A\text{-}III,\ B\text{-}I,\ C\text{-}IV,\ D\text{-}II
\]
\[
(C)\ A\text{-}I,\ B\text{-}IV,\ C\text{-}III,\ D\text{-}II
\]
\[
(D)\ A\text{-}II,\ B\text{-}III,\ C\text{-}I,\ D\text{-}IV
\]
the correct answer is:
\[
\boxed{A\text{-}IV,\ B\text{-}III,\ C\text{-}II,\ D\text{-}I}
\]
Hence:
\[
\boxed{\mathrm{(A)}}
\]