### Abstract

We study the design of small cost temporally connected graphs, under various constraints. We mainly consider undirected graphs of n vertices, where each edge has an associated set of discrete availability instances (labels). A journey from vertex u to vertex v is a path from u to v where successive path edges have strictly increasing labels. A graph is temporally connected iff there is a (u, v)-journey for any pair of vertices u, v, u ≠ v. We first give a simple polynomial-time algorithm to check whether a given temporal graph is temporally connected. We then consider the case in which a designer of temporal graphs can freely choose availability instances for all edges and aims for temporal connectivity with very small cost; the cost is the total number of availability instances used. We achieve this via a simple polynomial-time procedure which derives designs of cost linear in n. We also show that the above procedure is (almost) optimal when the underlying graph is a tree, by proving a lower bound on the cost for any tree. However, there are pragmatic cases where one is not free to design a temporally connected graph anew, but is instead given a temporal graph design with the claim that it is temporally connected, and wishes to make it more cost-efficient by removing labels without destroying temporal connectivity (redundant labels). Our main technical result is that computing the maximum number of redundant labels is APX-hard, i.e., there is no PTAS unless P = NP. On the positive side, we show that in dense graphs with random edge availabilities, there is asymptotically almost surely a very large number of redundant labels. A temporal design may, however, be minimal, i.e., no redundant labels exist. We show the existence of minimal temporal designs with at least nlogn labels.

Original language | English (US) |
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Pages (from-to) | 907-944 |

Number of pages | 38 |

Journal | Theory of Computing Systems |

Volume | 61 |

Issue number | 3 |

DOIs | |

State | Published - Oct 1 2017 |

### Keywords

- APX-hard
- Minimal graph
- Network design
- Random input
- Temporal graphs
- Temporally connected

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computational Theory and Mathematics

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## Cite this

*Theory of Computing Systems*,

*61*(3), 907-944. https://doi.org/10.1007/s00224-017-9757-x