The equivalence classes of graphs induced by the unsigned versions of the Reidemeister moves and delta-wye moves on knot diagrams are considered. Any graph that is reducible to a graph with no edges by some finite sequence of the unsigned version of the Reidemeister graph moves is called a knot graph. It is shown that the class of knot graphs strictly includes the class of delta-wye graphs. A relationship between the span of the bracket polynomial of a knot and \({\mathcal K}\)-equivalence, where \({\mathcal K}\) denotes the class of knot graphs is shown. It is also proved that the dimension of the intersection of the cycle and cocycle spaces is an effective numerical invariant of these classes.