Label-connected graphs and the gossip problem

F. Gobel, J. Orestes cerdeira, H.J. Veldman

Research output: Contribution to journalArticleAcademicpeer-review

12 Citations (Scopus)
65 Downloads (Pure)

Abstract

A graph with m edges is called label-connected if the edges can be labeled with real numbers in such a way that, for every pair (u, v) of vertices, there is a (u, v)-path with ascending labels. The minimum number of edges of a label-connected graph on n vertices equals the minimum number of calls in the gossip problem for n persons, which is known to be 2n − 4 for n ≥ 4. A polynomial characterization of label-connected graphs with n vertices and 2n − 4 edges is obtained. For a graph G, let θ(G) denote the minimum number of edges that have to be added to E(G) in order to create a graph with two edge-disjoint spanning trees. It is shown that for a graph G to be label-connected, θ(G) ≤ 2 is necessary and θ(G) ≤ 1 is sufficient. For i = 1, 2, the condition θ(G) ≤ i can be checked in polynomial time. Yet recognizing label-connected graphs is an NP-complete problem. This is established by first showing that the following problem is NP-complete: Given a graph G and two vertices u and v of G, does there exist a (u, v)-path P in G such that G−E(P) is connected?
Original languageEnglish
Pages (from-to)29-40
Number of pages12
JournalDiscrete mathematics
Volume0
Issue number87
DOIs
Publication statusPublished - 1991

Keywords

  • METIS-140352
  • IR-72986

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