Result: Constructive properties appear in some problems of graph theory and algorithms

Let \(\{\) x,y\(\}\) be any pair of independent vertices of a graph G. We describe a new method to determine the maximal circumference of G by observing max\(\{\) d(x),d(y)\(\}\). This is an extension of Ore's condition. In the second part of the paper we give some algorithms to obtain the shortest arborescences and optimal binary trees.