Ideals and graphs, Gröbner bases and decision procedures in graphs

The well known correspondence between even cycles of an undirected graph and polynomials in a binomial ideal associated to a graph is extended to odd cycles and polynomials in another binomial ideal. Other binomial ideals associated to an undirected graph are also introduced. The results about them with topics on monomial ideals are used in order to show decision procedures for bipartite graphs, minimal vertex covers, cliques, edge covers and matchings with algebraic tools. All such procedures are implemented in Maple 9.5.