A generalization of the pentomino exclusion problem: Dislocation of graphs

In this paper, we first investigate the pentomino exclusion problem, due to Golomb. We solve this problem on the 5xn grid and we give some lower and upper bounds for the kxn grid for all k and n. We then give a generalization of this problem in graphs, the @D-dislocation problem, which consists in finding the minimum number of vertices to be removed from a graph so as all the remaining connected components have cardinality at most @D. We investigate the algorithmic aspects of the @D-dislocation problem: we first prove the problem is NP-Complete, then we give a sublinear algorithm which solves the problem on a restricted class of graphs which includes the kxn grid graphs, provided k is not part of the input.