On graphs containing a given graph as center

Abstract

We examine the problem of embedding a graph H as the center of a supergraph G, and we consider what properties one can restrict G to have. Letting A(H) denote the smallest difference I V(G)I -I V(H)I over graphs G having center isomorphic to H it is demonstrated that A(H)54 for all H, and for 0 5 i 5 4 we characterize the class of trees T withA(T) = i. For n 2 2 and any graph H, we demonstrate a graph G with point and edge connectivity equal to n , with chromatic number x(G) = n 4-x(H), and whose center is isomorphic to H. Finally, if I V(H) I 2 9 and k 2 I V(H) I + 1, then for n sufficiently large (with n even when k is odd) we can construct a k-regular graph on n vertices whose center is isomorphic to H.

Eccentric graphs

“…It is well known (see [2]) that every graph is the center of some graph. In our final result, we characterize those graphs that are the center of some eccentric graph.…”

Closed 3-stop center and periphery in graphs

“…The closed 3-stop center C 3 (G) of G is the subgraph induced by those vertices of G having minimum closed 3-stop eccentricity [2]. For a given graph G, if there exists a graph H such that C 3 (H) ∼ = G, we define the closed 3-stop central appendage number of a graph G, AC 3 (G), to be the minimum difference |V (H)| − |V (G)| over all graphs H such that C 3 (H) ∼ = G. For more on standard central appendage number of a graph we refer the reader to [5].…”

Distance in stratified graphs

“…Hedetniemi (see [2]) showed that every graph is the center of some connected graph. Using this proof technique we establish the corresponding result for stratified graphs.…”