Line digraph iterations and connectivity analysis of de Bruijn and Kautz graphs

A graph has spread (m, k, l) if for any m+1 distinct nodes x, y₁, . . ., y_m and m positive integers r₁ , . . ., r_m, such that Σ_ir_i=k, there exist k node-disjoint paths of length at most 1 from x to the y_i, where r_i of them end at y_i. This concept contains, and is related to many important concepts used in communications and graph theory. The authors prove an optimal general theorem about the spreads of digraphs generated by line digraph iterations. Useful graphs, like the de Bruijn and Kautz digraphs, can be thus generated. The theorem is applied to the de Bruijn and Kautz digraphs to derive optimal bounds on their spreads, which implies previous results and resolves open questions on their connectivity, diameter, k-diameter, vulnerability, and some other measures related to length-bound disjoint paths