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The competition-independence game with preventionBrešar, Boštjan ; Mesarič Štesl, DašaNeodvisno dominacijsko igro, kot sta jo vpeljala Phillips in Slater, igrata na grafu dva igralca, Zmanjševalec in Napihovalec, ki si izmenjujeta poteze, v katerih lahko izbereta poljubno vozlišče, ki ... ni v zaprti soseščini kateregakoli od predhodno izbranih vozlišč. Cilj Zmanjševalca je ob koncu igre minimizirati (neodvisno) množico izbranih vozlišč, Napihovalčev cilj pa je ravno nasproten. Ob predpostavki, da oba igralca igrata optimalno glede na izbrana cilja, se porodita dve grafovski invarianti glede na to, kdo začne igro. V tem članku vpeljemo variacijo opisane igre, v kateri igralca lahko vsakič uporabita alternativno obliko poteze, pri kateri lahko označita (in ne izbereta!) poljubno vozlišče ▫$x$▫, ki še ni bilo odigrano, s čimer preprečita, da bi ▫$x$▫ bil izbran kadarkoli do konca igre. Med drugim to tudi pomeni, da ▫$x$▫ ne bo v končni množici izbranih vozlišč. Za dani graf ▫$G$▫ in ob predpostavki, da oba igralca igrata optimalno glede na svoja cilja, označimo s ▫$\widetilde{I}_{\rm d}(G)$▫ (oziroma ▫$\widetilde{I}_{\rm s}(G)$▫) moč množice izbranih vozlišč v neodvisni igri s preprečevanjem, ob pogoju da Zmanjševalec (oziroma Napihovalec) začne igro. Z uporabo Particijske strategije dokažemo, da za vsako naravno število ▫$n$▫ velja ▫$\widetilde{I}_{\rm d}(P_n)=\lfloor\frac{2n+3}{6}\rfloor$▫ in ▫$\widetilde{I}_{\rm s}(P_n)=\lfloor\frac{2n+4}{6}\rfloor$▫. Medtem ko ni težko ugotoviti splošnih mej, ▫$1\le \widetilde{I}_{\rm d}(G)\le \lfloor\frac{n}{2}\rfloor$▫ in ▫$1\le \widetilde{I}_{\rm s}(G)\le \lceil\frac{n}{2}\rceil$▫, v članku okarakteriziramo družine grafov, pri katerih v vsaki posamezni zgornji ali spodnji meji velja enakost. Nazadnje ugotovimo še tesno povezavo nove igre z variacijo igre barvanja, ki se imenuje pakirna igra barvanja, na grafih s premerom ▫$2$▫, zastavimo pa tudi več odprtih problemov.Source: Filomat. - ISSN 0354-5180 (Vol. 36, no. 18, 2022, str. 6197-6213)Type of material - article, component partPublish date - 2022Language - englishCOBISS.SI-ID - 137082627
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Brešar, Boštjan | 17005 |
Mesarič Štesl, Daša | 51477 |
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