The $e$-positivity of two families of $(claw, 2K_2)$-free graphs

Motivated by Stanley's conjecture about the $e$-positivity of claw-free incomparability graphs, Hamel and her collaborators studied the $e$-positivity of $(claw, H)$-free graphs, where $H$ is a four-vertex graph. In this paper we establish the $e$-positivity of generalized pyramid graphs and $2K_2$-free unit interval graphs, which are two important families of $(claw, 2K_2)$-free graphs. Hence we affirmatively solve one problem proposed by Hamel, Ho\`{a}ng and Tuero, and another problem considered by Foley, Ho\`{a}ng and Merkel.