For an arbitrary undirected simple graph G with m edges, we give an algorithm with running time O(m^4 |L|^2) to generate the set L of all minimal edge dominating sets of G. For bipartite graphs we obtain a better result; we show that their minimal edge dominating sets can be enumerated in time O(m^4 |L|). In fact our results are stronger; both algorithms generate the next minimal edge dominating set with incremental-polynomial delay O(m^5 |L|) and O(m^4 |L|) respectively, when L is the set of already generated minimal edge dominating sets. Our algorithms are tailored for and solve the equivalent problems of enumerating minimal (vertex) dominating sets of line graphs and line graphs of bipartite graphs, with incremental-polynomial delay, and consequently in output-polynomial time. Enumeration of minimal dominating sets in graphs has very recently been shown to be equivalent to enumeration of minimal transversals in hypergraphs. The question whether the minimal transversals of a hypergraph can be enumerated in output-polynomial time is a fundamental and challenging question in Output-Sensitive Enumeration; it has been open for several decades and has triggered extensive research in the field.
Polynomial Delay Algorithm for Listing Minimal Edge Dominating sets in Graphs
在输出多项式时间内，即在时间多项式内，计算超图的所有最小断面，即超图的最小断面的长度和最小断面的累积长度，是一个有50年历史的公开问题，迄今为止，很少有超图类的例子能解决这个问题.一个图的最小支配集是它的顶点集的一个子集，它与每个顶点的闭邻域都有一个非空的交。文[M. M. Kanté，V. Limouzy，A.玛丽湖Nourine，On the Enumeration of Minimal Dominating Sets and Related Notions，In Revision 2014]指出图中最小控制集的计数和超图中最小断面的计数是两个等价的问题。希望这种等价性可以帮助在transmittance问题中获得新的见解，很自然地查看图类内部。在[Golovach et al. - ICALP 2013]和[Kanté et al. - ISAAC 2012]中独立地并使用不同的技术证明了图中的最小边控制集（即线图中的最小控制集）可以在增量输出多项式时间中枚举。我们提供了第一个多项式延迟和多项式空间算法，该算法列出了图中的所有最小边控制集，回答了[Golovach et al. - ICALP 2013]的一个公开问题。除了结果，我们希望所使用的技术，是一个著名的Berge算法的修改和线图的结构的强有力的使用的混合，是非常感兴趣的，可以用来得到新的输出-多项式时间算法。
