  We prove a general inclusion-exclusion relation for the extended chromatic
symmetric function of a weighted graph, which specializes to (extended)
$k$-deletion, and we give two methods to obtain numerous new bases from
weighted graphs for the algebra of symmetric functions.
  Moreover, we classify when two weighted paths have equal extended chromatic
symmetric functions by proving this is equivalent to the classification of
equal ribbon Schur functions. This latter classification is dependent on the
operation composition of compositions, which we generalize to composition of
graphs. We then apply our generalization to obtain infinitely many families of
weighted graphs whose members have equal extended chromatic symmetric
functions.
