论文标题
最小数量的最大独立集中的无双图图
The minimum number of maximal independent sets in twin-free graphs
论文作者
论文摘要
确定某些图形类别中最大独立集数量最大数量的问题可以追溯到米勒和穆勒的论文，以及1960年代的Erdős和Moser问题。由于诸如恒星之类的简单示例，最低限度总是被认为不那么有趣。在本文中，我们表明，当不限于双人图的图表时，问题变得有趣，那里没有两个顶点具有相同的开放式邻居。我们考虑了任意图，两部分图和树的问题。最小独立集的最小数量在任意图的顶点数量，两部分图和树的指数上是对数。在后一种情况下，最低和极端图是由Taletski \uı和Malyshev确定的，但我们提供了较短的证明。
The problem of determining the maximum number of maximal independent sets in certain graph classes dates back to a paper of Miller and Muller and a question of Erdős and Moser from the 1960s. The minimum was always considered to be less interesting due to simple examples such as stars. In this paper we show that the problem becomes interesting when restricted to twin-free graphs, where no two vertices have the same open neighbourhood. We consider the question for arbitrary graphs, bipartite graphs and trees. The minimum number of maximal independent sets turns out to be logarithmic in the number of vertices for arbitrary graphs, linear for bipartite graphs and exponential for trees. In the latter case, the minimum and the extremal graphs have been determined earlier by Taletski\uı and Malyshev, but we present a shorter proof.
