Edge Exploration of Temporal Graphs

We introduce a natural temporal analogue of Eulerian circuits and prove that, in contrast with the static case, it is NP-hard to determine whether a given temporal graph is temporally Eulerian even if strong restrictions are placed on the structure of the underlying graph and each edge is active at only three times. However, we do obtain an FPT-algorithm with respect to a new parameter called intervalmembership-width which restricts the times assigned to different edges. Our techniques also allow us to resolve two open questions of Akrida, Mertzios and Spirakis [CIAC 2019] concerning a related problem of exploring temporal stars. Furthermore, we introduce a vertex-variant of interval-membership-width (which can be arbitrarily larger than its edge-counterpart) and use it to obtain an FPT-time algorithm for a natural vertex-exploration problem that remains hard even when interval-membership-width is bounded.

[1]  Rolf Niedermeier,et al.  Computing Maximum Matchings in Temporal Graphs , 2019, STACS.

[2]  Dimitris Fotakis,et al.  On the Size and the Approximability of Minimum Temporally Connected Subgraphs , 2016, ICALP.

[3]  Rolf Niedermeier,et al.  Adapting the Bron–Kerbosch algorithm for enumerating maximal cliques in temporal graphs , 2016, Social Network Analysis and Mining.

[4]  Ana Silva,et al.  Königsberg Sightseeing: Eulerian Walks in Temporal Graphs , 2021, IWOCA.

[5]  Jari Saramäki,et al.  Temporal Networks , 2011, Encyclopedia of Social Network Analysis and Mining.

[6]  Michal Pilipczuk,et al.  Parameterized Algorithms , 2015, Springer International Publishing.

[7]  Arnaud Casteigts,et al.  Finding Temporal Paths Under Waiting Time Constraints , 2021, Algorithmica.

[8]  Philipp Zschoche,et al.  Temporal Reachability Minimization: Delaying vs. Deleting , 2021, MFCS.

[9]  Afonso Ferreira,et al.  Computing Shortest, Fastest, and Foremost Journeys in Dynamic Networks , 2003, Int. J. Found. Comput. Sci..

[10]  Thomas Erlebach,et al.  On Temporal Graph Exploration , 2015, ICALP.

[11]  Paul G. Spirakis,et al.  Traveling salesman problems in temporal graphs , 2014, Theor. Comput. Sci..

[12]  Othon Michail.  An Introduction to Temporal Graphs: An Algorithmic Perspective* , 2016, Internet Math..

[13]  Paul G. Spirakis,et al.  The Complexity of Optimal Design of Temporally Connected Graphs , 2015, Theory of Computing Systems.

[14]  Frank Harary,et al.  Graph Theory , 2016 .

[15]  Paul G. Spirakis,et al.  How fast can we reach a target vertex in stochastic temporal graphs? , 2020, J. Comput. Syst. Sci..

[16]  Paul G. Spirakis,et al.  How fast can we reach a target vertex in stochastic temporal graphs? , 2019, ICALP.

[17]  Paul G. Spirakis,et al.  The temporal explorer who returns to the base , 2018, CIAC.

[18]  Amit Kumar,et al.  Connectivity and inference problems for temporal networks , 2000, Symposium on the Theory of Computing.

[19]  Jessica Enright,et al.  Deleting edges to restrict the size of an epidemic in temporal networks , 2018, MFCS.

[20]  Rolf Niedermeier,et al.  As Time Goes By: Reflections on Treewidth for Temporal Graphs , 2020, Treewidth, Kernels, and Algorithms.

[21]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[22]  Afonso Ferreira,et al.  Complexity of Connected Components in Evolving Graphs and the Computation of Multicast Trees in Dynamic Networks , 2003, ADHOC-NOW.

[23]  Paul G. Spirakis,et al.  Temporal Network Optimization Subject to Connectivity Constraints , 2013, Algorithmica.

[24]  Nicola Santoro,et al.  Time-varying graphs and dynamic networks , 2010, Int. J. Parallel Emergent Distributed Syst..

[25]  E-53: Solutio problematis ad geometriam situs pertinentis , 2020, Spectrum.

[26]  Othon Michail,et al.  An Introduction to Temporal Graphs: An Algorithmic Perspective* , 2015, Internet Math..

[27]  Amit Kumar,et al.  Connectivity and inference problems for temporal networks , 2000, STOC '00.

[28]  Hejun Wu,et al.  Efficient Algorithms for Temporal Path Computation , 2016, IEEE Transactions on Knowledge and Data Engineering.