Achievements of CORSO
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چکیده
The problem of interest is to determine the minimum number of edges of a 2-connected graph having a diameter equal to p. This problem deals with telecommunication survivable networks design with constraint of grade of service. In this article we prove the bounds of the number of edges for 2-connected graphs or for 2-edges-connected graphs with a diameter p. The bound for 2-connected graphs was a conjecture by B. Bollobás. One way to protect networks when some failure arises is to provide a sufficient connectivity to the network. In view of the robustness of the equipments and links 2-connectivity or 2-edgeconnectivity is sufficient. Notice also that k-connectivity, k ≥ 3 will lead to build too expensive networks. Another important parameter of a telecommunication network is the delay of transmission (or crossing delay). In first approximation, keeping the crossing delay under reasonnable values corresponds to bound the diameter of the network, and the cost of the network depends on the number of links. So we are interested in finding the minimum number of edges of a 2-connected (2-edge-connected) graph of given diameter p. In this work, we determine exactly these bounds, answering an old conjecture of the litterature due to B. Bollobás, see [11], he conjectured that the graphs obtained by linking two nodes by q chains with length p are extremal graphs for the class of 2-connected graphs with diameter p. Theorem Let G = (V,E) be an undirected 2-connected graph of diameter p ≥ 2. Then |E| ≥ d |V |p−(2p+1) p−1 e.
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