Minimum Temporal Spanners in Happy Graphs
Arnaud Casteigts, Hendrik Molter, Meirav Zehavi

TL;DR
This paper proves NP-hardness of finding minimum temporal spanners in happy graphs, a restricted class of temporal graphs, and explores parameterized complexity results.
Contribution
It establishes NP-hardness for minimum temporal spanners even in simple, proper graphs, and provides the first polynomial-time solution for graphs with a constant-size vertex cover.
Findings
NP-hardness holds even for simple, proper (happy) graphs.
Polynomial-time algorithm for graphs with constant-size vertex cover.
W[1]-hardness in non-happy graphs parameterized by feedback vertex number.
Abstract
Temporal graphs have edge sets that change over discrete time steps. Such graphs are temporally connected (TC) if all pairs of vertices can reach each other using paths that traverse the edges in a time-respecting way (temporal paths). Given a TC temporal graph it, a natural question is to find a minimum spanning subgraph of it that preserves temporal connectivity. These structures, known as temporal spanners, are fundamental and their properties (especially size) have been studied thoroughly in the past decade. In particular, the problem of minimizing the size of a temporal spanner is known to be hard. However, the existing results establish hardness for several incomparable settings and versions of the problem. In this article, we unify and strengthen these results by showing that this problem is NP-hard even on temporal graphs that are simple and proper (also known as "happy"),…
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