Domination game played on trees and spanning subgraphs

Boštjan Brešar (Author), Sandi Klavžar (Author), Douglas F. Rall (Author)

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 obstajajo 3-povezani grafi ▫$G$▫, ki vsebujejo 2-povezani vpeti podgraf ▫$H$▫, tako da je igralno dominacijsko število grafa ▫$H$▫ poljubno manjše od igralnega dominacijskega števila grafa ▫$G$▫. Podobno je dokazano, da za vsako celo število ▫$\ell \ge 1$▫ obstajata graf ▫$G$▫ in njegov vpeti podgraf $T$, tako da velja ▫$\gamma_g(G)-\gamma_g(T) \ge \ell$▫. Po drugi strani obstajajo grafi ▫$G$▫, za katere je igralno dominacijsko število vsakega vpetega drevesa v ▫$G$▫ poljubno večje od igralnega dominacijskega števila od ▫$G$▫.

Keywords

igra dominacije;igralno dominacijsko število;drevo;vpeti podgraf;graph theory;domination game;game domination number;tree;spanning subgraph;

Data

Language:	English
Year of publishing:	2013
Typology:	1.01 - Original Scientific Article
Organization:	UL FMF - Faculty of Mathematics and Physics
UDC:	519.17:519.83
COBISS:	16564313
ISSN:	0012-365X
Views:	43
Downloads:	25
Average score:	0 (0 votes)
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Other data

Secondary language:	Slovenian
Secondary title:	Igra dominacije na drevesih in vpetih podgrafih
Secondary 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 there exist 3-connected graphs ▫$G$▫ having a 2-connected spanning subgraph ▫$H$▫ such that the game domination number of ▫$H$▫ is arbitrarily smaller than that of ▫$G$▫. Similarly, for any integer ▫$\ell \ge 1$▫, there exists a graph ▫$G$▫ and a spanning tree ▫$T$▫ such that ▫$\gamma_g(G)-\gamma_g(T) \ge \ell$▫. On the other hand, there exist graphs ▫$G$▫ such that the game domination number of any spanning tree of ▫$G$▫ is arbitrarily larger than that of ▫$G$▫.
Secondary keywords:	igra dominacije;igralno dominacijsko število;drevo;vpeti podgraf;
URN:	URN:SI:UM:
Type (COBISS):	Not categorized
Pages:	str. 915-923
Volume:	Vol. 313
Issue:	iss. 8
Chronology:	2013
ID:	1476744