Generalizations of graphical parameters

Abstract

Graph theoretic parameters are often defined for a finite graph G with the aid of certain subsets of the vertex set V(G) or of the edge set E(G). One usually restricts attention to those subsets having maximum or minimum cardinality. Examples include the parameters associated with covering sets, matching sets, and separating sets. However, it may be desirable to consider a wider range of values, and we indicate how this can be accomplished for the vertex and edge versions of the parameters just mentioned.

Another generalization employs certain subsets of the union of V(G) and E(G), leading to such parameters as the total matching numbers and total connectivity numbers. The exact values of these parameters can be found for certain special classes of graphs and used to establish sharp upper and lower bounds for the corresponding parameters.

As a final example, one may generalize the concept of the genus of a graph by considering all compact orientable 2-manifolds in which a connected graph G has a 2-cell imbedding, and not only those for which the genus is a minimum.