20.500.12556/DiRROS-30350 Graphs with unique Grundy dominating sets Given a graph $G$ consider a procedure of building a dominating set $D$ in $G$ by adding vertices to $D$ one at a time in such a way that whenever vertex $x$ is added to $D$ there exists a vertex $y\in N_G[x]$ that becomes dominated only after $x$ is added to $D$. The maximum cardinality of a set $D$ obtained in the described way is called the Grundy domination number of $G$ and $D$ a Grundy dominating set. While a Grundy dominating set of a connected graph $G$ is not unique unless $G$ is the trivial graph, we consider a natural weaker uniqueness condition, notably that for every two Grundy dominating sets in a graph $G$ there is an automorphism that maps one to the other. We investigate both versions of uniqueness for several concepts of Grundy domination, which appeared in the context of domination games and are also closely related to zero forcing. For each of the four variations of Grundy domination we characterize the graphs that have only one Grundy dominating set of the given type, and characterize those forests that enjoy the weaker (isomorphism based) condition of uniqueness. The latter characterizations lead to efficient algorithms for recognizing the corresponding classes of forests. Grundy total domination number Grundy domination number zero forcing number trees graph automorphism Grundyjevo celotno dominacijsko število Grundyjevo dominacijsko število število ničelne prisile drevesa avtomorfizem grafa true false false Angleški jezik Slovenski jezik Neznano 2026-06-22 11:14:21 2026-06-22 11:14:21 2026-06-23 03:45:54 0000-00-00 00:00:00 2026 0 0 20 str. iss. 10, article no. 444 Vol. 45 Dec. 2026 0000-00-00 Zaloznikova Objavljeno NiDoloceno 0000-00-00 0000-00-00 2026-12-01 519.17 2238-3603 10.1007/s40314-026-03835-w 282411011 s40314-026-03835-w.pdf s40314-026-03835-w.pdf 1 53C31C2235052FCEC0F49EC3030776E8 892e3934b571a23f4cb8699aec272dc4ea8d2d8e74a60083e6be20dd9c4945fb 57a513ca-6e1b-11f1-98c5-001a4af901a5 https://dirros.openscience.si/Dokument.php?lang=slv&id=46046 https://link.springer.com/article/10.1007/s40314-026-03835-w 1 c371a852-6e1a-11f1-98c5-001a4af901a5 https://dirros.openscience.si/Dokument.php?lang=slv&id=46045 Inštitut za matematiko, fiziko in mehaniko 0 0 0