Define a minimal detour subgraph of the n-dimensional cube to be a spanning subgraph G of Qn having the property that for vertices x, y of Qn, distances are related by dG(x, y) ≤ dQn(x, y) + 2. Let f(n) be the minimum number of edges of such a subgraph of Qn. After preliminary work on distances in subgraphs of product graphs, we show that f(n)/|E(Qn| < c/√n. The subgraphs we construct to establish this bound have the property that the longest distances are the same as in Qn, and thus the diameter does not increase. We establish a lower bound for f(n), show that vertices of high degree must be distributed throughout a minimal detour subgraph of Qn, and end with conjectures and questions.
|Number of pages||10|
|Journal||Journal of Graph Theory|
|Publication status||Published - Oct 1996|
ASJC Scopus subject areas
- Geometry and Topology