In this paper, we study a type of incremental optimization variant of the Maximum Independent Set problem (MaxIS), called Bounded-Deletion Maximum Independent Set problem (BD-MaxIS): Given an unweighted graph $$G = (V, E)$$ , an initial feasible solution (i.e., an independent set) $$S^0\subseteq V$$ , and a non-negative integer k, the objective of BD-MaxIS is to find an independent set $$S\subseteq V$$ such that $$|S^0\setminus S|\le k$$ and |S| is maximized. The original MaxIS is generally NP-hard, but, it can be solved in polynomial time for perfect graphs (and therefore, comparability, co-comparability, bipartite, chordal, and interval graphs). In this paper, we show that BD-MaxIS is NP-hard even if the input is restricted to bipartite graphs, and hence to comparability graphs. On the other hand, fortunately, BD-MaxIS on co-comparability, interval, convex bipartite, and chordal graphs can be solved in polynomial time. Finally, we study the computational complexity on very similar variants of the Minimum Vertex Cover and the Maximum Clique problems for graph subclasses.

We study two variants of Maximum Cut, which we call Connected Maximum Cut and Maximum Minimal Cut, in this paper. In these problems, given an unweighted graph, the goal is to compute a maximum cut satisfying some connectivity requirements. Both problems are known to be NP-complete even on planar graphs whereas Maximum Cut on planar graphs is solvable in polynomial time. We first show that these problems are NP-complete even on planar bipartite graphs and split graphs. Then we give parameterized algorithms using graph parameters such as clique-width, tree-width, and twin-cover number. Finally, we obtain FPT algorithms with respect to the solution size.