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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: 591; Downloads: 11
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Abstract: Disertacija obravnava koordinacijo blagovnih tokov v maloprodajnih oziroma trgovinskih oskrbovalnih verigah. Zanje je značilno, da delujejo v nenehno spreminjajočem se okolju ter so močno odvisne od povpraševanja končnih odjemalcev. Tipično maloprodajno oskrbovalno verigo sestavlja več trgovin, ki so oskrbovane iz vsaj enega skladišča izbranega trgovca. Da dosežemo usklajenost zalog med objekti dane oskrbovalne verige, je zagotovo najprej potrebno doseči primerno koordinacijo vseh povezanih in odvisnih si aktivnosti, ki spremljajo blagovni tok. Pri gradnji matematičnega modela smo se najprej osredotočili na analizo povpraševanja, medtem ko so dinamično prilagajanje cen, upoštevanje pokvarljivosti itd. predmet nadaljnih raziskav, povezanih s komplementarnim modelom za operativno planiranje blagovnega toka. Idejno je pristop zasnovan tako, da se najprej poišče primerni distribucijski plan (taktični plan) z minimalnimi pričakovanimi stroški distribucije, kot tudi z minimalnimi pričakovanimi stroški rizikov. Problem predstavljamo kot večkriterijski problem optimizacije, predlagani stohastični model za dosego koordinacije pa pri tem kombinira več metod in postopkov iz področij teorije grafov, teorije verjetnosti in operacijskih raziskav. Ker je izbrani problem računsko kompleksen, za njegovo reševanje uporabimo nekaj hevrističnih postopkov lokalnega iskanja ter tudi predlagamo robustni optimizacijski pristop reševanja danega problema. Eksperimentalna študija pokaže, da je za optimizacijski problem mogoče implementirati algoritem, ki da uporabne rešitve.
Keywords: Distribucija, Zaloge, Optimizacija, Hevristike, Stohastično programiranje
Published in DKUM: 01.10.2019; Views: 1913; Downloads: 145
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Abstract: Roman domination is a historically inspired variety of general domination such that every vertex is labeled with labels from $\{0,1,2\}$. Roman domination number is the smallest of the sums of labels fulfilling condition that every vertex, labeled 0, has a neighbor, labeled 2. Using algebraic approach we give ▫$O(C)$▫ time algorithm for computing Roman domination number of special classes of polygraphs (rota- and fasciagraphs). By implementing the algorithm we give formulas for Roman domination number of the Cartesian products of paths and cycles ▫$P_n \Box P_k$▫, ▫$P_n \Box C_k$▫ for ▫$k \leq 8$▫ and ▫$n \in {\mathbb N}$▫ and for ▫$C_n \Box P_k$▫ and ▫$C_n \Box C_k$▫ for ▫$k \leq 5$▫, ▫$n \in {\mathbb N}$▫. We also give a list of Roman graphs among investigated families.
Keywords: graph theory, Roman domination number, Cartesian product, polygraphs, path algebra
Published in DKUM: 23.08.2017; Views: 1621; Downloads: 244
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Abstract: As links and nodes of interconnection networks are exposed to failures, one of the most important features of a practical networks design is fault tolerance. Vulnerability measures of communication networks are discussed including the connectivities, fault diameters, and measures based on Hosoya- Wiener polynomial. An upper bound for the edge fault diameter of product graphs is proved.
Keywords: discrete optimization, communication network, vulnerability
Published in DKUM: 21.07.2017; Views: 1109; Downloads: 108
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Abstract: Matching polynomial and perfect matchings for fasciagraphs, rotagraphs and twisted rotagraphs are treated in the paper. Classical transfer matrix approach makes it possible to get recursions for matching polynomial and perfect matchings, but the order of the matrix grows exponentially in the number of the linking edges between monographs. Novel transfer matrices are introduced whose order is much lower than that in classical transfer matrices. The virtue of the method introduced is especially pronounced when twoor more linking edges end in the same terminal vertex of a monograph. An example of a polyacene polygraph with extended pairings is given where a novel matrix has only 16 entries as compared to 65536 entries in the classical transfer matrix. However, all pairings are treated here on equal footing, but the method introduced can be applied to selected types of pairings of interest in chemistry.
Keywords: polygraphs, matching polynomial, matchings, perfect matchings, Kekulé structures, extended structures, recursive enumeration, transfer matrix method
Published in DKUM: 05.07.2017; Views: 1577; Downloads: 90
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