A cubic vertex kernel for Diamond-free Edge Deletion and more

A diamond is a graph obtained by removing an edge from a complete graph on four vertices. A graph is diamond-free if it does not contain an induced diamond. The Diamond-free Edge Deletion problem asks whether there exist at most $k$ edges in the input graph $G$ whose deletion results in a diamond-free graph. For this problem, a polynomial kernel of $O(k^4$) vertices was found by Fellows et. al. (Discrete Optimization, 2011). In this paper, we give an improved kernel of $O(k^3)$ vertices for Diamond-free Edge Deletion. Further, we give an $O(k^2)$ vertex kernel for a related problem {Diamond,Kt}-free Edge Deletion, where $t\geq 4$ is any fixed integer. To complement our results, we prove that these problems are NP-complete even for $K4$-free graphs and can be solved neither in subexponential time (i.e., $2^{o(|G|)}$) nor in parameterized subexponential time (i.e., $2^{o(k)}\cdot |G|^{O(1)}$), unless Exponential Time Hypothesis fails. Our reduction implies the hardness and lower bound for a general class of problems, where these problems come as a special case.