Problem of existence and construction of matchings in various classes of graphs is one of the well-known and important directions of research carried out in Discrete Mathematics. The subject stems from the prominent “Marriage Problem” which is a problem of existence and construction of a system of distinct representatives, for which necessary and sufficient condition is found. The interest towards problems which are related to matchings is explained not only by their theoretical importance, but also by the essential role that matchings play in the various, sometimes unexpected applications of Combinatorial Mathematics in development of methods for solving various practical problems. In the present work the influence of a maximum matching of the given graph on the graph obtained by its removal is investigated. The work can be interesting to researchers who work especially with matchings and related structures.