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

摘要

A graph has spread (m, k, l) if for any m+1 distinct nodes x, y/sub 1/, . . ., y/sub m/ and m positive integers r/sub 1/, . . ., r/sub m/, such that Sigma /sub i/r/sub i/=k, there exist k node-disjoint paths of length at most 1 from x to the y/sub i/, where r/sub i/ of them end at y/sub 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.