Browsing by Author "Zelinka, Bohdan"

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    Distances Between Graphs (Extended Abstract)
    (Technical university of Liberec, Czech Republic, 1992-01-01) Zelinka, Bohdan
    This chapter describes distances between isomorphism classes or distances between graphs. An isomorphism class of graphs is the class of all graphs that are isomorphic to a given graph. Two graphs whose distance is zero need not be identical but are isomorphic. A self-complementary graph is a graph that is isomorphic to its own complement. These graphs were studied independently by G. Ringel and H. Saclis. For the number n of vertices of a self-complementary graph, n 0 (mod 4) or n 1 (mod 4) always holds. An almost self-complementary graph can be defined as a graph that is isomorphic to a graph obtained from its complement by adding or deleting one edge.
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    Domatic Number of a Graph and its Variants (Extended Abstract)
    (Elsevier Science Bv, 2015-01-01) Adolph, Christoph H.; Akhunzyanov, R.; Alexeev, Maxim G.; Alexeev, G.D.; Amoroso, A.; Andrieux, Vincent; Anosov, V.; Austregesilo, Alexander; Badełek, Barbara; Balestra, Ferruccio; Barth, Jcl.; Baum, Günter; Beck, Reinhard; Bedfer, Yann; Berlin, Asher; Bernhard, Johannes; Bicker, K.; Bielert, Erwin; Bieling, Jakob; Birsa, Renato; Bisplinghoff, Jens; Bodlak, Martin; Böer, Marie; Bordalo, Paula; Bradamante, Franco; Braun, Christopher; Bressan, Andrea; Büchele, M.; Burtin, Etienne; Capozza, Luigi; Chiosso, Michela; Chung, Suhurk; Cicuttin, Andres; Crespo, María Liz; Curiel, Q.; Dalla Torre, Stefano; Dasgupta, Shivaji; Dasgupta, Subinay; Denisov, Oleg Yu; Donskov, Sergey; Doshita, Norihiro; Duic, Venicio; Dünnweber, Wolfgang; Dziewiecki, Michał; Efremov, Anatoli Vasilievich; Elia, C.; Eversheim, Paul Dieter; Eyrich, Wolfgang; Faessler, Martin; Ferrero, Andrea; Finger, Mir; Finger, Mic; Fischer, H.; Franco, Celso; du Fresne von Hohenesche, N.; Friedrich, Jan Michael; Frolov, Vladimir; Gautheron, F.; Gavrichtchouk, O.P.; Gerassimov, S.; Geyer, Reiner; Gnesi, I.; Gobbo, Benigno; Goertz, S.; Gorzellik, M.; Grabmüller, Stefanie; Grasso, Antonino; Grube, Boris; Grussenmeyer, Thomas; Guskov, Alexey; Haas, Florian; von Harrach, Dietrich; Hahne, Devin; Hashimoto, Ryo; Heinsius, F.H.; Herrmann, Felix; Hinterberger, Frank; Höppner, Christian; Horikawa, Naoaki; d'Hose, Nicole; Huber, Stefan; Ishimoto, Shigeru; Ivanov, Andrey; Ivanshin, Yu I.; Iwata, Takashi; Jahn, R.; Jary, Vladimír; Jasinski, Prometeusz Kryspin; Jörg, P.; Joosten, Rainer; Kabuß, Eva Maria; Ketzer, Bernhard; Khaustov, Guennady V.; Khokhlov, Yu A.; Kisselev, Yu F.; Klein, Frank J.; Klimaszewski, Krzysztof S.; Koivuniemi, Jaakko H.; Kolosov, Vladimir N.; Kondo, K.; Königsmann, Kay C.; Konorov, Igor A.; Konstantinov, V.F.; Kotzinian, Aram M.; Kouznetsov, Oleg L.; Krämer, Markus F.; Kroumchtein, Z.V.; Kuchinski, N.; Kunne, Fabienne; Kurek, Krzysztof; Kurjata, Robert P.; Lednev, Anatoly A.; Lehmann, Albert; Levillain, M.; Levorato, Stefano; Lichtenstadt, Jechiel; Maggiora, Angelo; Magnon, Alain; Makke, Nour; Mallot, Gerhard K.; Marchand, Claude; Martin, Anna D.; Marzec, J.; Matoušek, Jindřich; Matsuda, H.; Matsuda, Takeshi; Meshcheryakov, G.; Meyer, Wayne E.; Michigami, T.; Mikhailov, Yu V.; Miyachi, Y.; Nagaytsev, A.; Nagel, Thiemo; Nerling, Frank; Neubert, Sebastian; Neyret, Damien P.; Nový, Josef; Nowak, Wolf Dieter; Nunes, A.S.; Olshevsky, Alexander G.; Orlov, I.; Ostrick, Michael; Panknin, Robert; Panzieri, D.; Parsamyan, Bakur; Paul, Stephan M.; Peshekhonov, Dmitry V.; Platchkov, Stephane K.; Pochodzalla, Jozef.; Polyakov, Vladimir A.; Pretz, J.; Quaresma, Márcia; Quintans, Catarina; Ramos, Saturio; Regali, C.; Reicherz, Gerhard; Rocco, E.; Rossiyskaya, Natalia S.; Ryabchikov, Dmitri I.; Rychter, Andrzej; Samoylenko, Vladimir D.; Sandacz, Andrzej; Sarkar, Subir; Savin, Igor A.; Sbrizzai, Giulio; Schiavon, Paolo G.; Schill, Christian; Schlüter, Tobias; Schmidt, Katharina H.; Schmieden, Hartmut; Schönning, Karin; Schopferer, S.; Schott, Matthias; Shevchenko, O. Yu; Silva, Laura; Sinha, L.; Sirtl, S.; Slunecka, Miloslav; Sosio, S.; Sozzi, Federica; Srnka, Aleš; Steiger, Lukáš; Stolarski, Marcin; Sulej, Robert; Suzuki, Hajime; Szabelski, Adam; Szameitat, T.; Sznajder, Paweł; Takekawa, S.; ter Wolbeek, J.; Tessaro, S.; Tessarotto, F.; Thibaud, F.; Uhl, Sebastian; Uman, Igor U.; Virius, Miroslav; Wang, Li; Weisrock, T.; Wilfert, M.; Windmolders, Roland; Wollny, Heiner; Zaremba, Krzysztof; Zavertyaev, M.; Zemlyanichkina, Elena; Ziembicki, Marcin; Zink, Armin; Šulc, Miroslav; Zelinka, Bohdan
    This chapter presents some numerical invariants of graphs that are related to the concept of domination—namely, the domatic number and its variants.. The word domatic was coined from the words dominating and chromatic in the same way as the word smog was composed from the words smoke and fog. This concept is a certain analogy of the chromatic number, but instead of independent sets, dominating sets are used in its definition. A subset D of the vertex set V(G) of an undirected graphs G is called dominating if for each x V(G) − D there exists a vertex yD adjacent to x. A domatic partition of G is a partition of V(G), all of whose classes are dominating sets in G. The maximum number of classes of a domatic partition of G is called the “domatic number” of G and denoted by d(G). R. Laskar and S. T. Hedetniemi have introduced the connected domatic number d, (G) of a graph G. It is the maximum number of classes of a partition of V(G) into dominating sets that induce connected subgraphs of G.
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    Domination in bipartite graphs and in their complements
    (Elsevier, 2015) Zelinka, Bohdan
    The domatic numbers of a graph G and of its complement G were studied by J. E. Dunbar, T. W. Haynes and M. A. Henning. They suggested four open problems. We will solve the following ones: Characterize bipartite graphs G having d(G) = d(Ḡ). Further, we will present a partial solution to the problem: Is it true that if G is a graph satisfying d(G) = d(Ḡ), then γ(G) = γ(Ḡ)? Finally, we prove an existence theorem concerning the total domatic number of a graph and of its complement.
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    Edge domination in graphs of cubes
    (Springer Heidelberg, 2002-01-01) Zelinka, Bohdan
    The signed edge domination number and the signed total edge domination number of a graph are considered; they are variants of the domination number and the total domination number. Some upper bounds for them are found in the case of the n-dimensional cube Qn.
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    On a problem concerning stratified graphs
    (Elsevier, 2014) Zelinka, Bohdan
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    Remarks on restrained domination and total restrained domination in graphs
    (Czech Society of Chemical Engineering, 2005-01-01) Košek, Ondřej; Vyskočil, Jaroslav; Jodas, Bořivoj; Zelinka, Bohdan
    The restrained domination number γ r(G) and the total restrained domination number γ t r (G) of a graph G were introduced recently by various authors as certain variants of the domination number γ(G) of (G). A well-known numerical invariant of a graph is the domatic number d(G) which is in a certain way related (and may be called dual) to γ(G). The paper tries to define analogous concepts also for the restrained domination and the total restrained domination and discusses the sense of such new definitions. © Mathematical Institute, Academy of Sciences of Czech Republic 2005.
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    Signed and minus domination in bipartite graphs
    (Springer Heidelberg, 2006-01-01) Zelinka, Bohdan
    The paper studies the signed domination number and the minus domination number of the complete bipartite graph K p, q. © Mathematical Institute, Academy of Sciences of Czech Republic 2006.
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    Signed domination numbers of directed graphs
    (Czechoslovak Mathematical Journal, 2005-01-01) Zelinka, Bohdan
    The concept of signed domination number of an undirected graph (introduced by J. E. Dunbar, S. T. Hedetniemi, M. A. Henning and P. J. Slater) is transferred to directed graphs. Exact values are found for particular types of tournaments. It is proved that for digraphs with a directed Hamiltonian cycle the signed domination number may be arbitrarily small. © Mathematical Institute, Academy of Sciences of Czech Republic 2005.
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    Signed total domination number of a graph
    (Technická Univerzita v Liberci, 2001-01-01) Zelinka, Bohdan
    The signed total domination number of a graph is a certain variant of the domination number. If v is a vertex of a graph G, then N(v) is its oper neighbourhood, i.e. the set of all vertices adjacent to v in G. A mapping f : V(G) → {-1, 1}, where V(G) is the vertex set of G, is called a signed total dominating function (STDF) on G, if ∑x∈N(v) f(x) ≥ 1 for each v ∈ V(G). The minimum of values ∑x∈V(G) f(x), taken over all STDF's of G, is called the signed total domination number of G and denoted by γst(G). A theorem stating lower bounds for γst(G) is stated for the case of regular graphs. The values of this number are found for complete graphs, circuits, complete bipartite graphs and graphs on n-side prisms. At the end it is proved that γst(G) is not bounded from below in general.
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    Some remarks on domination in cubic graphs
    (Technical university of Liberec, Czech Republic, 1996-01-01) Zelinka, Bohdan
    We study three recently introduced numerical invariants of graphs, namely, the signed domination number γs, the minus domination number γ- and the majority domination number γmaj. An upper bound for γs and lower bounds for γ- and γmaj are found, in terms of the order of the graph.