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The concepts of critical and cocritical radius edge-invariant graphs are introduced. We prove that every graph can be embedded as an induced subgraph of a critical or cocritical radius-edge-invariant graph. We show that every cocritical radius-edge-invariant graph of radius r ≥ 15 must have at least 3r+2 vertices.
The diameter of a graph is the maximal distance between two vertices of . A graph is said to be diameter-edge-invariant, if for all its edges, diameter-vertex-invariant, if for all its vertices and diameter-adding-invariant if for all edges of the complement of the edge set of . This paper describes some properties of such graphs and gives several existence results and bounds for parameters of diameter-invariant graphs.
The eccentricity of a vertex is defined as the distance to a farthest vertex from . The radius of a graph is defined as a . A graph is radius-edge-invariant if for every , radius-vertex-invariant if for every and radius-adding-invariant if for every . Such classes of graphs are studied in this paper.
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