SOME OF THE BASIC TERM'S  REALTED  TO GRAPH THEORY

1. VERTEX COVER :-
        vertex cover of a graph is a set of vertices such that each edge of the graph is incident to atleast  one vertex of the set.

     In the above figure the vertex in the red color are vertex cover.

2. Minimum Vertex cover :-
        Minimum vertex cover problem for genaral graph (except Bipartite Graph)  is a optimization   problem in computer science. Minimum vertex cover is a vertex cover of smallest possible size.
   

       I think you understand by the example of figure's. see the difference in the figure of vertex cover and minimum vertex cover.

3. MATCHING :-
        A Matching or (Independent set of edges) is a set of edges without common vertices.
                   
                          

        In the figure set of red edges are matching.

4.MAXIMAL MATCHING :-
        A Maximal matching is also a matching of the graph ,such that it is not a proper subset of   any other matching in the graph.
        A maximal matching has a property if add another new edge that is not in matching ,it is no   longer a matching.
                                      

5.MAXIMUM MATCHING :-
        A Maximum matching is a maximal matching that containg largest possible number of edges.  Every maximum matching is maximal but not every maximal matching is maximum.

6. PERFECT MATCHING :-
        A Perfect matching is the matching which matches all the vertices of the graph.

7. NEAR PERFECT MATCHING :-
        A Near perfect matching is one in which exactly one vertex is unmatched. This can only occur when the graph has odd number of  vertices,such a matching must be maximum.
8.EDGE COVER :-
        Edge cover is a set of edges ,such that every vertex of grpah is incident to atleast one edge of the  set.