A characterization of Z(m)-well-covered graphs of girth 6 or more

The class of Z(m)-well-covered graphs, those in which the cardinality of ev ery maximal independent subset of vertices is congruent to the same number module m, contains the well-covered graphs as well as parity graphs. Here w e consider such graphs, where there is no small cycle present and obtain a characterization for those of girth 6 or more. Weighted well-covered graphs of girth 7 or more are also characterized, and this result is used, in tur n, to characterize magic well-covered graphs of girth 7 or more. (C) 2000 J ohn Wiley & Sons, Inc.