Kozhevnikov P.A.

What is the minimum number of connections required to organize a wired communication network of 10 nodes, so that the communication remains possible between any two remaining nodes (even through a chain of other nodes) in the event of any two nodes failing?

#
Evaluation. To ensure that the connection is maintained even when any two nodes fail, it is necessary that at least three lines of communication enter each node (if node $A$ is connected to two nodes $B$ and $C$, then when nodes $B$ and $C$ fail, node $A$ becomes inaccessible). Thus, the total number of communication lines should be no less than $10 \cdot 3: 2=15$.

Example: the framework of a pentagonal prism (10 vertices - nodes, and edges - lines of communication). If two nodes on one pentagon - the base of the prism - fail, the connection will still be maintained through the other pentagon. If one node fails on each of the different pentagons, the connection will also be maintained.