Domination game played on trees and spanning subgraphs

Brešar, Boštjan; Klavžar, Sandi; Rall, Douglas F.

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Title:	Domination game played on trees and spanning subgraphs
Authors:	ID Brešar, Boštjan (Author) ID Klavžar, Sandi (Author) ID Rall, Douglas F. (Author)
Files:	http://www.imfm.si/preprinti/PDF/01162.pdf
Language:	English
Work type:	Not categorized
Organization:	FNM - Faculty of Natural Sciences and Mathematics
Abstract:	Igra dominacije na grafu ▫$G$▫ je bila vpeljana v [B. Brešar, S. Klavžar, D. F. Rall, Domination game and an imagination strategy, SIAM J. Discrete Math. 24 (2010) 979-991]. Dva igralca, Dominator in Zavlačevalec, drug za drugim izbirata po eno vozlišče grafa. Vsako izbrano vozlišče mora povečati množico vozlišč, ki so bila dominirana do tega trenutka igre. Oba igralca izbirata optimalno strategijo, pri čemer Dominator želi igro končati v najmanjšem možnem številu korakov, Zavlačevalec pa v največjem možnem številu korakov. Igralno dominacijsko število ▫$gamma_g(G)$▫ je število izbranih vozlišč v igri, kjer je Dominator prvi izbral vozlišče. Ustrezno invarianto, ko igro začne Zavlačevalec, označimo z ▫$gamma_g'(G)$▫. V članku sta obe igri proučevani na drevesih in vpetih podgrafih. Dokazana je spodnja meja za igralno dominacijsko število drevesa, ki je funkcija njegovega reda in maksimalne stopnje. Pokazano je, da je meja asimptotično optimalna. Dokazano je, da za vsak ▫$k$▫ obstaja drevo ▫$T$▫ z ▫$(gamma_g(T),gamma_g'(T)) = (k,k+1)$▫ in postavljena je domneva, da ne obstaja drevo z ▫$(gamma_g(T),gamma_g'(T)) = (k,k-1)$▫. Obravnavana je povezava med igralnim dominacijskim številom grafa in njegovimi vpetimi podgrafi. Dokazano je, da za vsako naravno število ▫$ell geq 1$▫ obstaja graf ▫$G$▫ z vpetim drevesom ▫$T$▫, tako da velja ▫$gamma_g(G)-gamma_g(T)ge ell$▫. Nadalje obstajajo 3-povezani grafi ▫$G$▫, ki imajo vpeta drevesa z igralnim dominacijskim številom poljubno manjšim od ▫$G$▫.
Keywords:	igra dominacije, igralno dominacijsko število, drevo, vpeti podgraf, graph theory, domination game, game domination number, tree, spanning subgraph
Year of publishing:	2011
Number of pages:	str. 1-14
Numbering:	Vol. 49, št. 1162
PID:	20.500.12556/DKUM-49395
ISSN:	2232-2094
UDC:	519.17:519.83
COBISS.SI-ID:	16027993
NUK URN:	URN:SI:UM:DK:7PFXGIJT
Publication date in DKUM:	10.07.2015
Views:	1729
Downloads:	128
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Categories:	Misc.
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Secondary language

Language:	English
Title:	Igra dominacije na drevesih in vpetih podgrafih
Abstract:	The domination game, played on a graph ▫$G$▫, was introduced in [B. Brešar, S. Klavžar, D. F. Rall, Domination game and an imagination strategy, SIAM J. Discrete Math. 24 (2010) 979--991]. Vertices are chosen, one at a time, by two players Dominator and Staller. Each chosen vertex must enlarge the set of vertices of ▫$G$▫ dominated to that point in the game. Both players use an optimal strategy-Dominator plays so as to end the game as quickly as possible, Staller plays in such a way that the game lasts as many steps as possible. The game domination number ▫$gamma_g(G)$▫ is the number of vertices chosen when Dominator starts the game and the Staller-start game domination number ▫$gamma_g'(G)$▫ when Staller starts the game. In this paper these two games are studied when played on trees and spanning subgraphs. A lower bound for the game domination number of a tree in terms of the order and maximum degree is proved and shown to be asymptotically tight. It is shown that for every ▫$k$▫, there is a tree ▫$T$▫ with ▫$(gamma_g(T),gamma_g'(T)) = (k,k+1)$▫ and conjectured that there is none with ▫$(gamma_g(T),gamma_g'(T)) = (k,k-1)$▫. A relation between the game domination number of a graph and its spanning subgraphs is considered. It is proved that for any integer ▫$ell geq 1$▫, there exists a graph ▫$G$▫ and its spanning tree ▫$T$▫ such that ▫$gamma_g(G)-gamma_g(T)ge ell$▫. Moreover, there exist 3-connected graphs ▫$G$▫ having a spanning subgraph such that the game domination number of the spanning subgraph is arbitrarily smaller than that of ▫$G$▫.

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