  Gross derived an $O(n^2)$-time algorithm to calculate the genus distribution
of a given cubic Halin graph.
  In this paper, with the help of overlap matrix, we get a recurrence relation
for the Euler-genus polynomial of cubic caterpillar-Halin graphs.
  Explicit formulas for the embeddings of cubic caterpillar-Halin graph into a
surface with Euler-genus 0, 1 and 2 are also obtained.
