Switchings, realizations, and interpolation theorems for graph parameters

Only finite simple graphs are considered in this paper. For the most part, our notation and terminology follows that of Bondy and Murty [4]. Let G = (V ,E) denote a graph with vertex set V = V(G) and edge set E = E(G). We will use the following notation and terminology for a typical graph G. Let V(G) = {v1,v2, . . . ,vn} and E(G) = {e1,e2, . . . ,em}. We use |S| to denote the cardinality of a set S and therefore we define n= |V | the order of G and m = |E| the size of G. To simplify writing, we write e = uv for the edge e that joins the vertex u to the vertex v. The degree of a vertex v of a graph G is defined as dG(v) = |{e ∈ E : e = uv for some u∈ V}|. The maximum degree of a graph G is usually denoted by ∆(G). If S ⊆ V(G), the graph G[S] is the subgraph induced by S in G. For a graph G and X ⊆ E(G), we denote by G−X the graph obtained from G by removing all edges in X . If X = {e}, we write G− e for G−{e}. For a graph G and X ⊆V(G), the graph G−X is the graph obtained from G by removing all vertices in X and all edges incident with vertices in X . For a graph G and X ⊆ E(G), we denote by G+X the graph obtained from G by adding all edges in X . If X = {e}, we simply write G+ e for G+ {e}. Two graphs G andH are disjoint ifV(G)∩V(H) =∅. For any two disjoint graphsG andH , we define G∪H (their union) by V(G∪H) =V(G)∪V(H) and E(G∪H) = E(G)∪E(H). We can extend this definition to a finite union of pairwise disjoint graphs, since the operation “∪” is associative. For a positive integer p and a graph G, pG is denoted for the union of p copies of G. A graph G is said to be r-regular if all of its vertices have degree r. A 3-regular graph is called a cubic graph. Let G be a graph of order n and let V(G) = {v1,v2, . . . ,vn} be the vertex set of G. The sequence (dG(v1),dG(v2), . . . ,dG(vn)) is called a degree sequence of G. A sequence d = (d1,d2, . . . ,dn) of nonnegative integers is a graphic degree sequence if it is a degree sequence of some graph G. In this case, G is called a realization of d.