304 KB14 KB200820252015Yero, Ismael G.številka:1letnik:9str. 19-25COBISSID:17569369ISSN:1855-3966URN:URN:NBN:SI:doc-U7TFO1K9enUniverza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologijeArs mathematica contemporaneaCartesian product graphdominacijadominationkartezični produktni grafkrepki produktni grafparcialni produkt grafovpartial product of graphsstrong product graphVizingova domnevaVizing's conjecturePartial product of graphs and Vizing's conjecture|Let ?$G$? and ?$H$? be two graphs with vertex sets ?$V_1 = \{u_1, \dots, u_{n_1}\}$? and ?$V_2 = \{v_1, \dots, u_{n_2}\}$?, respectively. If ?$S \subset V_2$?, then the partial Cartesian product of ?$G$? and ?$H$? with respect to ?$S$? is the graph ?$G \Box_SH = (V,E)$?, where ?$V = V1 \times V_2$? and two vertices ?$(u_i, v_j)$? and ?$(u_k,v_j)$? are adjacent in ?$G \Box_SH$? if and only if either ?$(u_i = u_k \text{ and } v_j \sim v_l)$? or ?$(u_i \sim u_k \text{ and } v_j = v_l \in S)$?. If ?$A \subset V_1$? and ?$B \subset V_2$?, then the restricted partial strong product of ?$G$? and ?$H$? with respect to ?$A$? and ?$B$? is the graph ?$G_A \boxtimes_B H = (V,E)$?, where ?$V = V_1 \times V_2$? and two vertices ?$(u_i, v_j)$? and ?$(u_k,v_l)$? are adjacent in ?$G_A \boxtimes_B H$? if and only if either ?$(u_i = u_k \text{ and } v_j \sim v_l)$? or ?$(u_i \sim u_k \text{ and } v_j = v_l)$? or ?$(u_i \in A, u_k \notin A, v_j \in B, v_l \notin B, u_i \sim u_k \text{ and } v_j \sim v_l)$? or ?$(u_i \notin A, u_k \in A, v_j \notin B, v_l \in B, u_i \sim u_k \text{ and } v_j \sim v_l)$?. In this article we obtain Vizing-like results for the domination number and the independence domination number of the partial Cartesian product of graphs. Moreover we study the domination number of the restricted partial strong product of graphsNaj bosta ?$G$? in ?$H$? dva grafa z množicama vozlišč ?$V_1 = \{u_1, \dots, u_{n_1}\}$ in $V_2 = \{v_1, \dots, u_{n_2}\}$?. Če je ?$S \subset V_2$?, potem je parcialni kartezični produkt grafov ?$G$? in ?$H$? glede na ?$S$? graf ?$G \Box_SH = (V,E)$?, kjer je ?$V = V1 \times V_2$? in dve vozlišči ?$(u_i, v_j)$? in ?$(u_k,v_j)$? sta sosedni v ?$G \Box_SH$? če in samo če je bodisi ?$(u_i = u_k \text{ in } v_j \sim v_l)$? ali ?$(u_i \sim u_k \text{ in } v_j = v_l \in S)$?. Če je ?$A \subset V_1$ in $B \subset V_2$?, potem je zoženi parcialni krepki produkt grafov ?$G$? in ?$H$? glede na ?$A$? in ?$B$? graf ?$G_A \boxtimes_B H = (V,E)$?, kjer je ?$V = V_1 \times V_2$? in dve vozlišči ?$(u_i, v_j)$? and ?$(u_k,v_l)$? sta sosedni v ?$G_A \boxtimes_B H$? če in samo če je bodisi ?$(u_i = u_k \text{ in } v_j \sim v_l)$? bodisi? $(u_i \sim u_k \text{ in } v_j = v_l)$? ali ?$(u_i \in A, u_k \notin A, v_j \in B, v_l \notin B, u_i \sim u_k \text{ in } v_j \sim v_l)$? ali ?$(u_i \notin A, u_k \in A, v_j \notin B, v_l \in B, u_i \sim u_k \text{ in } v_j \sim v_l)$?. V članku dobimo Vizingovim podobne rezultate za dominacijsko število parcialnega kartezičnega produkta grafov. Poleg tega študiramo dominacijsko število zoženega parcialnega krepkega produkta grafovTEXTznanstveno časopisjejournalsSlovenian National E-content AggregatorNational and University Library of SloveniaUniverza na Primorskem, Fakulteta za naravoslovje, matematiko in informacijske tehnologije