On central-peripheral appendage numbers of uniform central graphs

Abstract

In a uniform central graph (UCG) the set of eccentric vertices of a central vertex is the same for all central vertices. This collection of eccentric vertices is the centered periphery. For a pair of graphs ( C , P ) the central-peripheral appendage number, A ucg ( C , P ), is the minimum number vertices needed to be adjoined to the graphs C and P in order to construct a uniform central graph H with center V ( C ) and centered-periphery V ( P ). We compute A ucg ( C , P ) in terms of the radius and diameter of P and whether or not C is a complete graph. In the process we show A ucg ( C , P ) ≤ 6 if diam( P ) > 2. We also provide structure theorems for UCGs in terms of the centered periphery.