Trapezoid graphs and their coloring
Discrete Applied Mathematics
A linear algorithm for bipartition of biconnected graphs
Information Processing Letters
Most uniform path partitioning and its use in image processing
Discrete Applied Mathematics - Special issue: combinatorial structures and algorithms
On the structure of trapezoid graphs
Discrete Applied Mathematics
Computational Statistics & Data Analysis
Approximating the maximally balanced connected partition problem in graphs
Information Processing Letters
On the approximability of some maximum spanning tree problems
Theoretical Computer Science - Special issue: Latin American theoretical informatics
A linear-time algorithm for four-partitioning four-connected planar graphs
Information Processing Letters
Journal of the ACM (JACM)
An efficient algorithm to solve connectivity problem on trapezoid graphs
Journal of Applied Mathematics and Computing
Graph Theory
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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.