Geodesic
A pursuit-evasion differential game with slow pursuers on the edge graph of simplexes.~I
Matematičeskaâ teoriâ igr i eë priloženiâ, Tome 12 (2020) no. 4, pp. 7-23.

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We consider the differential game between several pursuing points and one evading point moving along the graph of edges of a simplex when maximal quantities of velocities are given. The normalization of the game in the sense of J. von Neumann including the description of classes of admissible strategies is exposed. In the present part of the paper the qualitative problem for the full graph of three dimensional simplex is solved using the strategy of parallel pursuit for a slower pursuer and some numerical coefficient of a simplex characterizing its proximity to the regular one. Next part will be devoted to higher dimensional cases.
Keywords: differential game, game on a graph, pursuit problem, evasion problem, П-strategy, coefficient of regularity of a simplex, full graph. 
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Abdulla A. Azamov; Tolanbay T. Ibaydullayev. A pursuit-evasion differential game with slow pursuers on the edge graph of simplexes.~I. Matematičeskaâ teoriâ igr i eë priloženiâ, Tome 12 (2020) no. 4, pp. 7-23. http://geodesic.mathdoc.fr/item/MGTA_2020_12_4_a1/

[1] A. A. Azamov, “Ob alternative dlya differentsialnykh igr presledovaniya-ubeganiya na beskonechnom intervale vremeni”, Prikl. mat. i mekh., 50:4 (1986), 561–566 | MR | Zbl

[2] A. Azamov, “Nizhnyaya otsenka koeffitsienta preimuschestva v zadache poiska na grafakh”, Differents. uravneniya, 44:12 (2008), 1764–1767 | MR | Zbl

[3] A. A. Azamov, A. Sh. Kuchkarov, A. G. Kholboev, “Igra presledovaniya-ubeganiya na rebernom ostove pravilnykh mnogogrannikov. I; II; III”, MTIiP, 7:3 (2015), 3–15 ; 8:4 (2016), 3–13 ; 11:4 (2019), 5–23 | MR | Zbl | MR | Zbl | Zbl

[4] M. A. Bulgakova, L. A. Petrosyan, “Mnogoshagovye igry s poparnym vzaimodeistviem na polnom grafe”, MTIiP, 11:1 (2019), 3–20 | MR | Zbl

[5] Dzh. Kelli, Obschaya topologiya, Fizmatgiz, M., 1968

[6] N. N. Krasovskii, A. I. Subbotin, Pozitsionnye differentsialnye igry, Nauka, M., 1976 | MR

[7] N. N. Petrov, Teoriya igr, Izd-vo UdmGU, Izhevsk, 1997

[8] L. A. Petrosyan, Differentsialnye igry presledovaniya, Izd-vo LGU, L., 1977

[9] L. A. Petrosyan, A. A. Sedakov, “Mnogoshagovye setevye igry s polnoi informatsiei”, MTIiP, 1:2 (2009), 66–81 | Zbl

[10] L. A. Petrosyan, N. A. Zenkevich, E. V. Shevkoplyas, Teoriya igr, BKhV-Peterburg, SPb, 2012

[11] L. S. Pontryagin, “K teorii differentsialnykh igr”, Uspekhi mat. nauk, 21:4 (1966), 219–272 | MR

[12] B. N. Pshenichnyi, V. V. Ostapenko, Differentsialnye igry, Naukova dumka, Kiev, 1992 | MR

[13] T. Andreae, “Note on a pursuit game played on graphs”, Discrete Appl. Math., 9:2 (1984), 111–115 | DOI | MR | Zbl

[14] T. Andreae, “A search problem on graphs which generalizes some group testing problems with two defectives”, Combinatorics of ordered sets (Oberwolfach, 1988), Discrete Math., 88, no. 2–3, 1991, 121–127 | DOI | MR | Zbl

[15] T. Andreae, “A two-person game on graphs where each player tries to encircle his opponent's men”, Theoret. Comput. Sci., 215:1–2 (1999), 305–323 | DOI | MR | Zbl

[16] A. A. Azamov, B. Samatov, $\Pi$-strategy. An elementary introduction to the theory of differential games, National Univ. of Uzbekistan, 2000

[17] C. Berge, “Colloque sur la Theorie des Jeux” (Tenu a Bruxelles le 29 et le 30 mai 1975), Cahiers Centre Etudes Recherche Oper., 18:1-2 (1976), 1–253 (French) | MR

[18] A. Bonato, P. Golovach, G. Hahn, J. Kratochvl, “The capture time of a graph”, Discrete Mathematics, 309:18 (2009), 5588–5595 | DOI | MR | Zbl

[19] A. Bonato, R. J. Nowakowski, The game of cops and robbers on graphs, Student Mathematical Library, 61, American Mathematical Society, Providence, RI, 2011 | DOI | MR | Zbl

[20] F. V. Fomin, D. M. Thilikos, “An annotated bibliography on guaranteed graph searching”, Theoret. Comput. Sci., 399 (2008), 236–245 | DOI | MR | Zbl

[21] A. Friedman, Differential games, Wiley, NY, 1971 | MR | Zbl

[22] T. Gavenc̆iak, “Cop-win graphs with maximum capture-time”, Discrete Mathematics, 310:10–11 (2010), 1557–1563 | MR | Zbl

[23] P. A. Golovach, N. N. Petrov, F. V. Fomin, “Search in graphs”, Proc. Steklov Inst. Math., 2000, no. Suppl. 1, Control in Dynamic Systems, 90–103 | MR

[24] R. Isaacs, Differential games, Dover Publications, 1999 | Zbl

[25] B. Kummer, Spiele auf Graphen, International Series of Numerical Mathematics, 44, Birkhauser Verlag, Basel–Boston, Mass., 1980 (German) | MR | Zbl

[26] R. J. Nowakowski, “Unsolved problems in combinatorial games”, Games of no chance, v. 5, Math. Sci. Res. Inst. Publ., 70, Cambridge Univ. Press, Cambridge, 2019, 125–168 | MR | Zbl

[27] W. Sierpinski, Cardinal, Ordinal Numbers, Warsaw Polish scientific publishers, 1965 | MR | Zbl