Radius-invariant graphs

Abstract

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Diameter-invariant graphs

“…there are at least two edge-disjoint x-y paths of length d in G. Thus there are no levels i, i + 1 both with only one vertex. Because of this we have at most 1 2 d + 1 levels with only one vertex if d is even and at most 1 2 (d + 1) levels with only one vertex if d is odd. Any vertex v on level i can be adjacent only to vertices on levels i − 1, i, i + 1.…”

“…(3) If the cycle F and the x-y geodesic have 1 2 d vertices in common, then a is adjacent to at most 4 vertices of the cycle F and the x-y geodesic together. If the cycle F and the x-y geodesic have 1 2 d − i vertices in common, then a is adjacent to at most 4 + i vertices of the cycle F and the x-y geodesic together. Otherwise d G+uv (x, y) < d(G).…”

“…From the practical point of view we need to study the stability of the radius and the diameter of a graph G, especially when an arbitrary edge or vertex is removed from G. This operation can represent a single failure of communication line or any communication center (processor, etc.). The papers [1], [3], [5], [13] examine several properties of graphs in which radii do not change under these two conditions, and moreover, when an arbitrary edge is added to the graph G. These graphs are defined as follows: Definition 1.1. A graph G is:…”