Longest path in a particular type of graph

I know that the longest path problem is NP-hard for a general graph. However, I am considering a particular kind of graph, consisting of one cycle, plus one additional edge incident on each vertex of the cycle. For example, for a cycle of length 7, we have the graph:

All the edges are weighted (the weight is a real number and can be positive or negative). I want to find the largest simple path on this graph, where the size of a path is the sum of the weights of the edges on the path.

The algorithm should be linear in the size of the cycle. But any ideas are appreciated.