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On critical and cocritical radius edge-invariant graphs

Ondrej Vacek — 2008

Discussiones Mathematicae Graph Theory

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.

Diameter-invariant graphs

Ondrej Vacek — 2005

Mathematica Bohemica

The diameter of a graph G is the maximal distance between two vertices of  G . A graph G is said to be diameter-edge-invariant, if d ( G - e ) = d ( G ) for all its edges, diameter-vertex-invariant, if d ( G - v ) = d ( G ) for all its vertices and diameter-adding-invariant if d ( G + e ) = d ( e ) for all edges of the complement of the edge set of G . This paper describes some properties of such graphs and gives several existence results and bounds for parameters of diameter-invariant graphs.

Radius-invariant graphs

Vojtech BálintOndrej Vacek — 2004

Mathematica Bohemica

The eccentricity e ( v ) of a vertex v is defined as the distance to a farthest vertex from v . The radius of a graph G is defined as a r ( G ) = min u V ( G ) { e ( u ) } . A graph G is radius-edge-invariant if r ( G - e ) = r ( G ) for every e E ( G ) , radius-vertex-invariant if r ( G - v ) = r ( G ) for every v V ( G ) and radius-adding-invariant if r ( G + e ) = r ( G ) for every e E ( G ¯ ) . Such classes of graphs are studied in this paper.

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