On the Parameterized Complexity of Maximum Degree Contraction Problem

In the Maximum Degree Contraction problem, input is a graph G on n vertices, and integers k, d, objective to check whether can be transformed into of maximum degree at most using k edge contractions. A simple brute-force algorithm that checks all possible sets edges for solution runs in time $$n^{\mathcal {O}(k)}$$ . As our first result, we prove this asymptotically optimal, upto constants exponents, under Exponential Time Hypothesis (ETH). Belmonte, Golovach, van’t Hof, Paulusma studied problem realm parameterized complexity proved, among other things, it admits an FPT running $$(d + k)^{2k} \cdot n^{\mathcal {O}(1)} = 2^{\mathcal {O}(k \log (k+d) )} {O}(1)}$$ , remains NP-hard every constant $$d \ge 2$$ (Acta Informatica (2014)). We present different $$2^{\mathcal {O}(dk)} particular, {O}(k)} fixed d. same article, authors asked polynomial kernel, when by $$k d$$ answer question negative does not admit compression unless $$\textsf {NP}\subseteq \textsf {coNP}/poly$$