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Abstract: We introduce a new setting for dealing with the problem of the domination number of the Cartesian product of graphs related to Vizing's conjecture. The new framework unifies two different approaches to the conjecture. The most common approach restricts one of the factors of the product to some class of graphs and proves the inequality of the conjecture then holds when the other factor is any graph. The other approach utilizes the so-called Clark-Suen partition for proving a weaker inequality that holds for all pairs of graphs. We demonstrate the strength of our framework by improving the bound of Clark and Suen as follows: ɣ(X◻Y) ≥ max{1/2ɣ(X) ɣt(Y), 1/2ɣt(X) ɣ(Y)}, where ɣ stands for the domination number, ɣt is the total domination number, and X◻Y is the Cartesian product of graphs X and Y.
Keywords: Cartesian product, total domination, Vizing's conjecture, Clark and Suen bound
Published in DKUM: 09.08.2024; Views: 70; Downloads: 3
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Abstract: ▫$2$▫-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.
Keywords: 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
Published in DKUM: 28.02.2024; Views: 247; Downloads: 5
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Abstract: Let γ(D) denote the domination number of a digraph D and let C$_m$□C$_n$ denote the Cartesian product of C$_m$ and C$_n$, the directed cycles of length n ≥ m ≥ 3. Liu et al. obtained the exact values of γ(C$_m$□C$_n$) for m up to 6 [Domination number of Cartesian products of directed cycles, Inform. Process. Lett. 111 (2010) 36–39]. Shao et al. determined the exact values of γ(C$_m$□C$_n$) for m = 6, 7 [On the domination number of Cartesian product of two directed cycles, Journal of Applied Mathematics, Volume 2013, Article ID 619695]. Mollard obtained the exact values of γ(C$_m$□C$_n$) for m = 3k + 2 [M. Mollard, On domination of Cartesian product of directed cycles: Results for certain equivalence classes of lengths, Discuss. Math. Graph Theory 33(2) (2013) 387–394.]. In this paper, we extend the current known results on C$_m$□C$_n$ with m up to 21. Moreover, the exact values of γ(C$_n$□C$_n$) with n up to 31 are determined.
Keywords: domination number, Cartesian product, directed cycle
Published in DKUM: 02.09.2022; Views: 585; Downloads: 11
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Abstract: The paper examines two ubiquitous concepts of power: the "classical sociological" concept which draws on Max Weber's definition of power, and the "Foucauldian" concept which stems from Michel Foucault's genealogical works. Three main theses are argued for. First, the two concepts are not, in most respects, as radically different as it is usually claimed. It is demonstrated that both can make room for different sources of power, for understanding power in a non-reified way, for the fact that power is rarely completely centralised, etc. Second, in those respects in which the two concepts actually differ, the classical view of power is more convincing and useful than the Foucauldian one. It is demonstrated that the Foucauldian view is implicitly positivist in the normative domain and thus unable to differentiate between power and domination, and that it succumbs to errors of methodological holism (i. e. undertheorising agency). Third, it is argued that the classical sociological view allows to analytically distinguish between power, domination and exploitation. These three categories are shown not to be synonymous and to carry with them importantly different sociological implications. It is demonstrated that exploitation cannot merely refer to any process of unpaid appropriation of surplus as obvious false positives are generated from this definition. Nonetheless, such appropriation is the fundamental characteristic which differentiates exploitation from domination (but not power itself), and this reveals an important sociological implication for the dynamics of struggle of the exploited against exploitation in contrast to the struggle of the dominated against the dominators.
Keywords: power, domination, exploitation, Foucault, value-neutrality
Published in DKUM: 24.10.2017; Views: 1495; Downloads: 164
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Abstract: A graph is an efficient open domination graph if there exists a subset of vertices whose open neighborhoods partition its vertex set. We characterize those graphs ▫$G$▫ for which the Cartesian product ▫$G \Box H$▫ is an efficient open domination graph when ▫$H$▫ is a complete graph of order at least 3 or a complete bipartite graph. The characterization is based on the existence of a certain type of weak partition of ▫$V(G)$▫. For the class of trees when ▫$H$▫ is complete of order at least 3, the characterization is constructive. In addition, a special type of efficient open domination graph is characterized among Cartesian products ▫$G \Box H$▫ when ▫$H$▫ is a 5-cycle or a 4-cycle.
Keywords: efficient open domination, Cartesian product, vertex labeling, total domination
Published in DKUM: 10.07.2017; Views: 1075; Downloads: 162
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