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Star partitions on graphs
Discrete Optimization ( IF 1.1 ) Pub Date : 2019-01-25 , DOI: 10.1016/j.disopt.2019.01.002
G. Andreatta , C. De Francesco , L. De Giovanni , P. Serafini

Given an undirected graph, a star partition is a partition of the nodes into subsets with at least two nodes so that the subgraph induced by each subset has a spanning star. Star partitions are related to well-known problems concerning domination in graphs and edge covering. We focus on the Constrained Star Partition Problem (CSP) that asks for finding a star partition of given cardinality. The problem is new and presents interesting peculiarities. We explore the relation between the cardinalities of star partitions and domatic bipartitions, showing that there are star partitions of any cardinality between minimum and maximum values, and that a similar but weaker result holds for domatic bipartitions. We study the computational complexity of different versions of star partition and domatic bipartition problems, proving that most of them, in particular CSP, constrained domatic bipartition and balanced domatic bipartition, are NP-complete. We also show that star partition problems are polynomial on trees and, more generally, on bounded treewidth graphs. We introduce an integer linear programming formulation that defines a polytope containing all the star partitions of a graph, showing that its vertices have only integral components for trees, which implies that linear programming can be used to solve weighted star partition problems on trees.



中文翻译:

图上的星形分区

在给定无向图的情况下,星形分区是将节点划分为具有至少两个节点的子集,以便由每个子集生成的子图具有生成星。星形分区与众所周知的有关图形和边缘覆盖中的控制的问题有关。我们关注于约束星分配问题(CSP),该问题要求查找给定基数的星分配。这个问题是新的,并且存在一些有趣的特性。我们探索了星形分区的基数和半球形划分之间的关​​系,表明在最小值和最大值之间存在任意基数的星形分区,并且类似的但较弱的结果适用于球形划分。我们研究了不同版本的恒星分区和半球分区问题的计算复杂性,证明了其中大多数,尤其是CSP,受约束的主体划分和平衡的主体划分是NP完全的。我们还表明,星形分配问题是树上的多项式,更常见的是有界树宽图上的多项式。我们引入了一个整数线性规划公式,该公式定义了包含图的所有星形分区的多面体,这表明其顶点仅具有树木的积分成分,这意味着线性规划可用于解决树木的加权星形分区问题。

更新日期:2019-01-25
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