Separating path systems
Graphs, Hypergraphs, and Computing
18 March 15:30 - 16:30
Victor Falgas-Ravry - Umeå University
Let G be a graph on n vertices. A family F of paths in G constitutes a separating system of G if for ever pair of distinct edges e,f in E(G) there exists a path p in F which contains exactly one of e and f. How small a separating path system can we find?
This question arises naturally in the context of network design. The graph G represents a communication network in which one link is defective; to identify this link, we can send messages between nodes along predetermined paths. If a message does not reach its destination, then we deduce that the defective link lies on the corresponding path. A minimal separating path system thus allows us to determine precisely which link is defective using a minimal number of messages.
We show that for asymptotically almost all n-vertex graphs, we can find a separating system containing at most 48 n paths. In addition we prove some exact extremal results in the case where G is a tree.
This is joint work with Teeradej Kittipassorn, Daniel Korandi, Shoham Letzter and Bhargav Narayanan. Similar results have recently and independently been obtained by Balogh, Csaba, Martin and Pluhar.
Magnus M. Halldorsson
Adam Mickiewicz University
Technical University of Denmark, DTU