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Telephone problems with failures. (English) Zbl 0578.05059
Summary: Consider a multigraph \(G\) on \(n\) vertices whose edges are linearly ordered. The vertices of \(G\) may represent people and the edges two-way communication between pairs of people. A vertex \(v\) is \(k\)-failure-safe in communicating with a vertex \(w\) if there is a path of ascending edges from \(v\) to \(w\) even when any \(k\)-edges of \(G\) are deleted. In this paper, we show that the minimum size \(\mu (n,k)\) of \(G\) such that one vertex communicates \(k\)-failure-safe with every other vertex is given by \(\mu (n,k)=\lceil ((k+2)/2)(n-1)\rceil\) for \(k\leq n-2\) and \(\mu (n,k)=\lceil ((k+1)/2n\rceil\) for \(k\geq n-2\). We also show that for \(k\geq 1\) the minimum size \(\tau (n,k)\) of \(G\) such that every vertex communicates \(k\)-failure-safe with every other vertex satisfies \(\mu (n,k)+n-2\lceil \sqrt{n}\rceil \leq \tau (n,k)\leq \lfloor (k+3/2)(n-1)\rfloor.\) The problem of finding \(\tau(n,k)\) for \(k=0\) is the well-known telephone problem.

MSC:
05C35 Extremal problems in graph theory
90B10 Deterministic network models in operations research
94C15 Applications of graph theory to circuits and networks
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