  We introduce path-conjoined graphs defined for two rooted graphs by joining
their roots with a path, and investigate the chromatic symmetric functions of
its two generalizations: spider-conjoined graphs and chain-conjoined graphs. By
using the composition method developed by Zhou and the third author, we obtain
neat positive $e_I$-expansions for the chromatic symmetric functions of
clique-path-cycle graphs, path-clique-path graphs, and path-clique-clique
graphs.
