Line digraph iterations and the spread concept—with application to graph theory, fault tolerance, and routing

Abstract

This paper is concerned with the spread concept, line digraph iterations, and their relationship. A graph has spread (m, k, l) if for any m + 1 distinct nodes x,y ₁,..., y _m and m positive integers r ₁,..., r _m such that ⌆_ir_i= k, there exist k node-disjoint paths of length at most l 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 line digraph of a digraph G(V, E) is the digraph L(G) where nodes represent the edges of G and there is an edge (x,y) in L(G) if and only if x represents the edge (u, v) in G and y represents the edge (v,w) in G for some u, v,w ε V(G). Many useful graphs, like the de Bruijn and Kautz digraphs, can be generated by line digraph iterations. We prove an optimal general theorem about the spreads of digraphs generated by line digraph iterations. Then we apply it to the de Bruijn and Kautz digraphs to derive optimal bounds on their spreads, which improve, re-prove, or resolve previous results and open questions on the connectivity, diameter, k-diameter, diameter vulnerability, and some other issues related to length-bounded disjoint paths, of these two graphs.

This research is supported by grant NSF-STC88-09648. On leave from Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing.