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A Game of Cops and Robbers on Graphs with Periodic Edge-Connectivity.

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conference contribution
posted on 21.05.2020, 08:13 by Thomas Erlebach, Jakob T Spooner

This paper considers a game in which a single cop and a single robber take turns moving along the edges of a given graph G. If there exists a strategy for the cop which enables it to be positioned at the same vertex as the robber eventually, then G is called cop-win, and robber-win otherwise. In contrast to previous work, we study this classical combinatorial game on edge-periodic graphs. These are graphs with an infinite lifetime comprised of discrete time steps such that each edge is assigned a bit pattern of length le, with a 1 in the i-th position of the pattern indicating the presence of edge in the i-th step of each consecutive block of le steps. Utilising the known framework of reach-ability games, we obtain an O(LCM(L)·n3) time algorithm to decide if a given n-vertex edge-periodic graph Gτ is cop-win or robber-win as well as compute a strategy for the winning player (here, L is the set of all edge pattern lengths le, and LCM(L) denotes the least common multiple of the set L). For the special case of edge-periodic cycles, we prove an upper bound of 2·l·LCM(L) on the minimum length required of any edge-periodic cycle to ensure that it is robber-win, where l= 1 if LCM(L)≥2·maxL, and l= 2 otherwise. Furthermore, we provide constructions of edge-periodic cycles that are cop-win and have length1.5·LCM(L) in the l= 1 case and length 3·LCM(L) in the l= 2 case.

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Citation

Erlebach T., Spooner J.T. (2020) A Game of Cops and Robbers on Graphs with Periodic Edge-Connectivity. In: Chatzigeorgiou A. et al. (eds) SOFSEM 2020: Theory and Practice of Computer Science. SOFSEM 2020. Lecture Notes in Computer Science, vol 12011. Springer, Cham

Source

International Conference on Current Trends in Theory and Practice of Informatics SOFSEM 2020

Version

AM (Accepted Manuscript)

Published in

Lecture Notes in Computer Science

Volume

12011

Pagination

64 - 75

Publisher

Springer Verlag (Germany)

issn

0302-9743

isbn

978-3-030-38918-5

Copyright date

2020

Editors

Chatzigeorgiou A; Dondi R; Herodotou H; Kapoutsis CA; Manolopoulos Y; Papadopoulos GA; Sikora F

Book series

Lecture Notes in Computer Science

Spatial coverage

Limassol, Cyprus

Temporal coverage: start date

20/01/2020

Temporal coverage: end date

24/01/2020

Language

en

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