FMF, Mathematical Library, Lj. (MAKLJ)
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Domination in digraphs and their direct and Cartesian productsBrešar, Boštjan ; Kuenzel, Kirsti ; Rall, Douglas F.Dominanantna (oz. celotno dominantna) množica ▫$S$▫ digrafa ▫$D$▫ je taka množica vozlišč digrafa, da je unija zaprtih (oz. odprtih) izhodnih okolic vozlišč iz ▫$S$▫ enaka množici vozlišč ▫$V(D)$▫ ... digrafa ▫$D$▫. Najmanjša moč dominantne (oz. celotno dominantne) množice v $D$ je dominantno (oz. celotno dominantno) število digrafa ▫$D$▫, kar označimo z ▫$\gamma(D)$▫ (oz. ▫$\gamma_t(D)$▫). Največje število paroma disjunktnih zaprtih (oz. odprtih) vhodnih okolic vozlišč iz ▫$D$▫ označimo z ▫$\rho(D)$▫ (oz. ▫$\rho^{o}(D)$▫). V članku dokažemo, da v digrafih, katerih temeljni grafi imajo ožino vsaj ▫$7$▫, zaprte (oz. odprte) vhodne okolice zadoščajo Hellyjevi lastnosti in ta dva rezultata uporabimo za dokaz, da v vsakem usmerjenem drevesu (to je, v digrafu, katerega temeljni graf je drevo) velja ▫$\gamma_t(T)=\rho^o(T)$▫ in ▫$\gamma(T)=\rho(T)$▫. Z uporabo prve enakosti nato dokažemo, da velja ▫$\gamma_t(G\times T)=\gamma_t(G)\gamma_t(T)$▫, kjer je ▫$G$▫ poljuben digraf in ▫$T$▫ poljubno usmerjeno drevo, kjer oba digrafa nimata izvora, in je ▫$G\times T$▫ njun direktni produkt. Iz enakosti ▫$\gamma(T)=\rho(T)$▫ pa izpeljemo mejo ▫$\gamma(G\Box T)\ge\gamma(G)\gamma(T)$▫, kjer je ▫$G$▫ poljuben digraf, ▫$T$▫ poljubno usmerjeno drevo in ▫$G\Box T$▫ njun kartezični produkt. V splošnih digrafih takšna meja Vizingovega tipa ne velja, zato pa dokažemo, da za poljubna digrafa ▫$G$▫ in ▫$H$▫, kjer je ▫$\gamma(G)\ge\gamma(H)$▫, velja ▫$\gamma(G \mathbin{\Box} H) \ge \frac{1}{2} \gamma(G)(\gamma(H) + 1)$▫. Ta neenakost je ostra, kar demonstriramo z neskončno družino primerov parov digrafov, za katere je dosežena enakost v tej neenakosti. V preostanku članka raziskujemo lastnosti usmerjenih dreves ▫$T$▫ in digrafov ▫$H$▫, ki dosežejo enakost ▫$\gamma(T\mathbin{\Box} H)=\gamma(T)\gamma(H)$▫.Source: Journal of graph theory. - ISSN 0364-9024 (Vol. 99, iss. 3, March 2022, str. 359-377)Type of material - article, component part ; adult, seriousPublish date - 2022Language - englishCOBISS.SI-ID - 94534659
Author
Brešar, Boštjan |
Kuenzel, Kirsti |
Rall, Douglas F.
Topics
usmerjeni graf |
dominacija |
pakiranje |
kartezični produkt |
direktni produkt |
Cartesian product |
digraph |
direct product |
domination |
packing
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Database name | Field | Year |
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Brešar, Boštjan | 17005 |
Kuenzel, Kirsti | |
Rall, Douglas F. |
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