# On Cops and Robbers on $G^{\Xi}$ and cop-edge critical graphs.

Keywords:
Cops and Robbers, vertex-pursuit games

### Abstract

Cop Robber game is a two player game played on an undirected graph. In this game cops try to capture a robber moving on the vertices of the graph. The cop number of a graph is the least number of cops needed to guarantee that the robber will be caught. In this paper we presents results concerning games on $G^{\Xi}$, that is the graph obtained by connecting the corresponding vertices in $G$ and its complement $\overline{G}$. In particular we show that for planar graphs $c(G^{\Xi})\leq 3$. Furthermore we investigate the cop-edge critical graphs, i.e. graphs that for any edge $e$ in $G$ we have either $c(G-e)<c(G) \text{ or } c(G-e)>c(G)$. We show couple examples of cop-edge critical graphs having cop number equal to~$3$.### References

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G. MacGillivray and K. Seyfarth, Domination numbers of planar graphs, J. Graph Theory 22 (1996), 213-229.

R. Nowakowski and P. Winkler, Vertex-to-vertex pursuit in a graph, Discrete Math. 43 (1983), 235-239.

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P. Pralat, When does a random graph have constant cop number?, Australian Journal of Combinatorics 46 (2010), 285-296.

A. Quilliot, Homomorphisms, points xes, retractions et jeux de poursuite dans les graphs, les ensembles ordonnes et les espaces metriques, Ph.D. thesis, Universite de Paris VI, 1983.

Z. A. Wagner, Cops and robbers on diameter two graphs, arXiv:1312.0755v3, 2014.

M. Aigner and M. Fromme, A game of cops and robbers, Discrete Appl. Math. 8 (1984), 1-11.

W. D. Baird, Cops and robbers on graphs and hypergraphs, 2011, Theses and dissertations. Paper 821.

W. D. Baird, A. Beveridge, P. Codenotti, A. Maurer, J. McCauley, and S. Valeva, On the minimum order of k-cop-win graphs, Contrib. Discrete Math. 9 (2014), 70-84.

H.-J. Bandelt and E. Prisner, Clique graphs and Helly graphs, J. Combin. Theory B 51 (1991), 34-45.

A. Berarducci and B. Intrigrila, On the cop number of a graph, Adv. Appl. Math. 14 (1993), 389-403.

A. Beveridge, A. Maurer, J. McCauley, and S. Valeva, Cops and rob-

bers on planar graphs, https://apps.carleton.edu/curricular/math/assets/

PursuitEvasionReport3.pdf, report.

A. Bonato and R. J. Nowakowski, The game of cops and robbers on graphs, Student Mathematical Library, vol. 61, American Mathematical Society, 2011.

A. Brandstadt, V. B. Le, and J. P. Spinrad, Graph classes : a survey, Monographs on Discrete Mathematics and Applications, vol. 3, SIAM, 1999.

N. E. Clarke, S. L. Fitzpatrick, A. Hill, and R. J. Nowakowski, Edge critical cops and robber, Discrete Math. 310 (2010), 2299-2309.

S. L. Fitzpatrick, Edge-critical cops and robber in planar graphs, Discrete Math. 329 (2014), 1-11.

W. Goddard and M. A. Henning, Domination in planar graphs with small diameter, J. Graph Theory 40 (2002), 1-25.

F. Jaeger and C. Payan, Relations du type Nordhais-Gaddum pour le nombre d'absorption d'un graphe simple, C. R. Acad. Sci. Paris S'er. A 274 (1972), 728-730.

J. P. Joseph and S. Arumugam, Domination in graphs, Internat. J. Management Systems 11 (1995), 177-182.

G. MacGillivray and K. Seyfarth, Domination numbers of planar graphs, J. Graph Theory 22 (1996), 213-229.

R. Nowakowski and P. Winkler, Vertex-to-vertex pursuit in a graph, Discrete Math. 43 (1983), 235-239.

T. Poston, Fuzzy geometry, Ph.D. thesis, University of Warwick, 1971.

P. Pralat, When does a random graph have constant cop number?, Australian Journal of Combinatorics 46 (2010), 285-296.

A. Quilliot, Homomorphisms, points xes, retractions et jeux de poursuite dans les graphs, les ensembles ordonnes et les espaces metriques, Ph.D. thesis, Universite de Paris VI, 1983.

Z. A. Wagner, Cops and robbers on diameter two graphs, arXiv:1312.0755v3, 2014.

Published

2017-11-27

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