1-join composition for $α$-critical graphs

Valencia, Carlos E.; Barrita, Marcos I.

1-join composition for $α$-critical graphs

Authors

Description

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.
18 pages, 7 figures