Reducing Graph Parameters by Contractions and Deletions

Abstract We consider the following problem: for a given graph G and two integers k d , can we apply fixed operation at most times in order to reduce parameter $$\pi $$ π by least ? show that this problem is NP-hard when independence number vertex deletion or edge contraction, even $$d=1$$ d = 1 restricted chordal graphs. give polynomial time algorithm bipartite graphs fixed. Further, complete complexity dichotomy H -free clique contraction showing $$(C_3+P_1)$$ ( C 3 + P ) . When chromatic number, determine computational of associated cographs multipartite Our results answer several open questions stated Diner et al. (Theor Comput Sci 746:49–72, 2012, https://doi.org/10.1016/j.tcs.2018.06.023 ).