1. On Grundy total domination number in product graphsBoštjan Brešar, Csilla Bujtás, Tanja Dravec, Sandi Klavžar, Gašper Košmrlj, Tilen Marc, Balázs Patkós, Zsolt Tuza, Máté Vizer, 2021, izvirni znanstveni članek Opis: A longest sequence (v1,....,vk) of vertices of a graph G is a Grundy total dominating sequence of G if for all i, N(vi)\U{j=1}^{i-1} N(vj)≠∅. The length k of the sequence is called the Grundy total domination number of G and denoted ɣ{gr}^{t}(G). In this paper, the Grundy total domination number is studied on four standard graph products. For the direct product we show that ɣ{gr}^{t}(G x H) > ɣ{gr}^{t}(G)ɣ{gr}^{t}(H), conjecture that the equality always holds, and prove the conjecture in several special cases. For the lexicographic product we express ɣ{gr}^{t}(G o H) in terms of related invariant of the factors and find some explicit formulas for it. For the strong product, lower bounds on ɣ{gr}^{t}(G ⊠ H) are proved as well as upper bounds for products of paths and cycles. For the Cartesian product we prove lower and upper bounds on the Grundy total domination number when factors are paths or cycles.
Ključne besede: total domination, Grundy total domination number, graph product
Objavljeno v DKUM: 07.08.2024; Ogledov: 96; Prenosov: 11
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2. Packings in bipartite prisms and hypercubesBoštjan Brešar, Sandi Klavžar, Douglas F. Rall, 2024, izvirni znanstveni članek Opis: ▫$2$▫-pakirno število ▫$\rho_2(G)$▫ grafa ▫$G$▫ je kardinalnost največjega ▫$2$▫-pakiranja grafa ▫$G$▫, odprto pakirno število ▫$\rho^{\rm o}(G)$▫ pa kardinalnost največjega odprtega pakiranja grafa ▫$G$▫, kjer je odprto pakiranje (oz. ▫$2$▫ pakiranje) množica vozlišč grafa ▫$G$▫, katerih dve (zaprti) soseščini se ne sekata. Dokazano je, da če je ▫$G$▫ dvodelen, potem je ▫$\rho^{\rm o}(G\Box K_2) = 2\rho_2(G)$▫. Za hiperkocke sta določeni spodnji meji ▫$\rho_2(Q_n) \ge 2^{n - \lfloor \log n\rfloor -1}$▫ in ▫$\rho^{\rm o}(Q_n) \ge 2^{n - \lfloor \log (n-1)\rfloor -1}$▫. Te ugotovitve so uporabljene za injektivna barvanja hiperkock. Dokazano je, da je ▫$Q_9$▫ najmanjša hiperkocka, ki ni popolno injektivno obarvljiva. Dokazano je tudi, da je ▫$\gamma_t(Q_{2^k}\times H) = 2^{2^k-k}\gamma_t(H)$▫, kjer je ▫$H$▫ poljuben graf brez izoliranih vozlišč.
Ključne besede: 2-pakirno število, odprto pakirno število, dvodelna prizma, hiperkocke, injektivno barvanje, celotno dominacijsko število, 2-packing number, open packing number, bipartite prism, hypercube, injective coloring, total domination number
Objavljeno v DKUM: 28.02.2024; Ogledov: 260; Prenosov: 11
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3. Lower bounds for domination and total domination number of direct products graphsGašper Mekiš, 2009 Opis: An exact lower bound for the domination number and the total domination number of the direct product of finitely many complete graphs is given: ▫$gamma(times_{i=1}^t K_{n_i} ge t+1$▫, ▫$t ge 3$▫. Sharpness is established in the case when the factors are large enough in comparison to the number of factors. The main result gives a lower bound for the domination (and the total domination) number of the direct product of two arbitrary graphs: ▫$gamma(G times H) ge gamma(G) + gamma(H) - 1$▫. Infinite families of graphs that attain the bound are presented. For these graphs it also holds ▫$gamma_t(G times H) = gamma(G) + gamma(H) - 1$▫. Some additional parallels with the total domination number are made.
Ključne besede: matematika, teorija grafov, dominacijska množica, dominacijsko število, celotna dominacijska množica, celotno dominacijsko število, direktni produkt grafov, poln graf, mathematics, graph theory, dominating set, domination number, total dominating set, total domination number, direct product graphs, complete graphs
Objavljeno v DKUM: 10.07.2015; Ogledov: 1305; Prenosov: 45
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4. Gravier, Sylvain: Total domination number of grid graphs. - Discrete Appl. Math. 121 (2002), no. 1-3, 119-128Sandi Klavžar, 2003, recenzija, prikaz knjige, kritika Ključne besede: matematika, teorija grafov, rešetke, dominantno število, mathematics, graph theory, grid graphs, total domination number
Objavljeno v DKUM: 10.07.2015; Ogledov: 1340; Prenosov: 35
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