Some special minimum [formula omitted]-geodetically connected graphs

Abstract

A connected graph G is k -geodetically connected ( k -GC) if the removal of less than k vertices does not affect the distances (lengths of the shortest paths) between any pair of the remaining vertices. As such graphs have important applications in robust system designs, we are interested in the minimum number of edges required to make a k -GC graph of order n , and characterizing those minimum k -GC graphs. When 3 < k < ( n − 1 ) / 2 , minimum k -GC graphs are not yet known in general, even the minimum number of edges m ( n , k ) is not determined. In this paper, we will determine all of the minimum k -GC graphs for an infinite set of special ( n , k ) pairs that were formerly unknown. To derive our results, we also developed new bounds on m ( n , k ) . Additionally, we show that k -GC graphs with small relative optimality gaps can be easily constructed and expanded with great flexibilities, which gives convenient applications for robust system designs.