1-$α$ -临界图并合成
1-join composition for $α$-critical graphs
摘要：
给定两个图G和H，其1-{ \it join}是取G和H的不交并，将G的一个非空子集顶点与H的一个非空子集顶点之间的所有边相加得到的图。图的合成运算在图论的一些结构性结果中起着重要的作用，特别是1-联合成在无爪图、无牛图、完美图等几类图的分解定理中起着重要的作用。一个图G称为{ \it $α$ -critical}，如果对G的所有边e，其中$α(G)$，G的{ \it稳定数}，等于G的独立集的最大基数，并且G的顶点集M是{ \it稳定}的，如果M中没有两个顶点相邻。对$α$ -临界图的研究是非常重要的，例如，对$α$ -临界图的完整描述可以很好地刻画G的稳定数.本文给出了G和H的1-联是$α$ -临界图所必须满足的充要条件。因此，我们得到了一个非常有用的方法来构造基本$α$ -临界图使用图的1-联。
英文摘要：
Given two graphs G and H its 1-{\it join} is the graph obtained by taking the disjoint union of G and H and adding all the edges between a nonempty subset of vertices of G and a nonempty subset of vertices of H. In general, composition operations of graphs has played a fundamental role in some structural results of graph theory and in particular the 1-join composition has played an important role in decomposition theorems of several class of graphs such as the claw-free graphs, the bull-free graphs, the perfect graphs, etc. A graph G is called {\it $α$-critical} if $α(G\setminus e)> α(G)$ for all the edges e of G, where $α(G)$, the {\it stability number} of G, is equal to the maximum cardinality of a stable set of G, and a set of vertices M of G is {\it stable} if no two vertices in M are adjacent. The study $α$-critical graphs is important, for instance a complete description of $α$-critical graphs would yield a good characterization of the stability number of G. In this paper we give necessary and sufficient conditions that G and H must satisfy in order to its 1-join will be an $α$-critical graph. Therefore we get a very useful way to construct basic $α$-critical graphs using the 1-join of graphs.
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