Dominacijska igra (Brešar in sod. v SIAM J Discrete Math 24:979-991, 2010) in celotna dominacijska igra (Henning in sod. v Graphs Comb 31:1453-1462. 2015) sta dobro znani igri na grafih za dva ...igralca - Dominatorja in Zavlačevalko. V tem članku vpeljemo tri nove različice dominacijske igre, in sicer Z-dominacijsko igro, L-dominacijsko igro in LL-dominacijsko igro. Za vsako izmed novih iger dokažemo princip nadaljevanja, ki nam med drugim pove, da se rezultata igre (pripadajaoči grafovski invarianti) v primerih, ko Dominator začne oziroma, ko začne Zavlačevalka, razlikujeta za največ ena. V članku vzpostavimo hierarhijo med petimi dominacijskimi igrami. Za vsako izmed novih treh iger določimo spodnjo in zgornjo mejo v odvisnosti od dominacijske ali celotne dominacijske igre ter reda grafa. Izračunali smo vrednosti novih treh invariant na poteh v odvisnosti od števila vozlišč do majhne konstante natančno. Na koncu smo našteli še nekaj odprtih problemo ter postavili domnevo, ki pravi, da L-igralno dominacijsko število ni nikoli večje kot 6/7 reda grafa.
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