Graph theory is increasingly commonly utilised in genetics, proteomics and neuroimaging. In such fields, the data of interest generally constitute weighted graphs. Analysis of such weighted graphs often require the integration of topological metrics with respect to the density of the graph. Here, density refers to the proportion of the number of edges present in that graph. When topological metrics based on shortest paths are of interest, such density-integration usually necessitates the iterative application of Dijkstra's algorithm in order to compute the shortest path matrix at each density level. In this short note, we describe a recursive shortest path algorithm based on single edge updating, which replaces the need for the iterative use of Dijkstra's algorithm. Our proposed procedure is based on pairs of breadth-first searches around each of the vertices incident to the edge added at each recursion. An algorithmic analysis of the proposed technique is provided. When the graph of interest is coded as an adjacency list, our algorithm can be shown to be more efficient than an iterative use of Dijkstra's algorithm.