Max-min weight balanced connected partition

For a connected graph and a positive integral vertex weight function , a max-min weight balanced connected -partition of , denoted as , is a partition of into disjoint vertex subsets such that each (the subgraph of induced by ) is connected, and is maximum. Such a problem has a lot of applications in image processing and clustering, and was proved to be NP-hard. In this paper, we study on a special class of graphs: trapezoid graphs whose maximum degree is bounded by a constant. A pseudo-polynomial time algorithm is given, based on which an FPTAS is obtained for . A step-stone for the analysis of the FPTAS depends on a lower bound for the optimal value of in terms of the total weight of the graph. In providing such a lower bound, a byproduct of this paper is that any 4-connected trapezoid graph on at least seven vertices has a 4-contractible edge, which may have a value in its own right.