Max-min weight balanced connected partition

For a connected graph $$G=(V,E)$$ and a positive integral vertex weight function $$w$$, a max-min weight balanced connected $$k$$-partition of $$G$$, denoted as $$BCP_k$$, is a partition of $$V$$ into $$k$$ disjoint vertex subsets $$(V_1,V_2,\ldots ,V_k)$$ such that each $$G[V_i]$$ (the subgraph of $$G$$ induced by $$V_i$$) is connected, and $$\min _{1\le i\le k}\{w(V_i)\}$$ 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 $$BCP_k$$ 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 $$k=2,3,4$$. A step-stone for the analysis of the FPTAS depends on a lower bound for the optimal value of $$BCP_k$$ 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.