Labelconnected graphs and the gossip problem
Göbel, F. and Orestes Cerdeira, J. and Veldman, H.J. (1991) Labelconnected graphs and the gossip problem. Discrete Mathematics, 87 (1). pp. 2940. ISSN 0012365X

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Abstract:  A graph with m edges is called labelconnected 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 labelconnected 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 labelconnected 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 edgedisjoint spanning trees. It is shown that for a graph G to be labelconnected, 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 labelconnected graphs is an NPcomplete problem. This is established by first showing that the following problem is NPcomplete: 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? 
Item Type:  Article 
Copyright:  © 1991 Elsevier 
Faculty:  Electrical Engineering, Mathematics and Computer Science (EEMCS) 
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Link to this item:  http://purl.utwente.nl/publications/72986 
Official URL:  https://doi.org/10.1016/0012365X(91)90068D 
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