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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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References:
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