Adjacent cycle-chains are $e$-positive

We describe a way to decompose the chromatic symmetric function as a positive sum of smaller pieces. We show that these pieces are $e$-positive for cycles. Then we prove that attaching a cycle to a graph preserves the $e$-positivity of these pieces. From this, we prove an $e$-positive formula for graphs of cycles connected at adjacent vertices. We extend these results to graphs formed by connecting a sequence of cycles and cliques.