Diameter-vital edges in a graph

The graph resulting from contracting edge e is denoted as G/e and the graph resulting from deleting edge e is denoted as G - e. An edge e is diameter-essential if diam(G/e) < diam(G), diameter-increasing if diam(G - e) < diam(G), and diameter-vital if it is both diameter-essential and diameter-increasing. We partition the edges that are not diameter-vital into three categories. In this paper, we study realizability questions relating to the number of edges that are not diameter-vital in the three defined categories. A graph is diameter-vital if all its edges are diameter-vital. We give a structural characterization of diameter-vital graphs. © 2011 Springer Basel AG