Rainbow domination in the lexicographic product of graphs

Tadeja Kraner Šumenjak (Author), Douglas F. Rall (Author), Aleksandra Tepeh (Author)

Abstract

Preslikava iz množice vozlišč grafa ▫$G$▫ v potenčno množico množice ▫$\{1,2,\dots, k\}$▫ se imenuje ▫$k$▫-mavrična dominantna funkcija, če za poljubno vozlišče ▫$v$▫ z lastnostjo ▫$f(v) = \emptyset$▫ velja ▫$\{1,\dots,k\} = \bigcup_{u \in N(v)}f(u)$▫. Obravnavamo ▫$k$▫-mavrično dominantno število grafa ▫$G$▫, ▫$\gamma_{rk}(G)$▫, ki je minimalna vsota (po vseh vozliščih grafa ▫$G$▫) moči podmnožic, ki so vozliščem dodeljena s ▫$k$▫-mavrično dominantno funkcijo. V članku se osredotočimo na 2-mavrično dominantno število leksikografskega produkta grafov in dokažemo natančno spodnjo in zgornjo mejo za to število. Dejansko pokažemo natančno vrednost za ▫$\gamma_{r2}(G \circ H)$▫, razen v primeru, ko je ▫$\gamma_{r2}(H) = 3$▫ in obstaja taka minimalna 2-mavrična dominantna funkcija grafa $H$, ki nekemu vozlišču v grafu ▫$H$▫ dodeli oznako ▫$\{1,2\}$▫.

Keywords

dominacija;popolna dominacija;mavrična dominacija;leksikografski produkt;domination;total domination;rainbow domination;lexicographic product;

Data

Language:	English
Year of publishing:	2013
Typology:	1.01 - Original Scientific Article
Organization:	UM FERI - Faculty of Electrical Engineering and Computer Science
UDC:	519.17
COBISS:	3514668
ISSN:	0166-218X
Views:	883
Downloads:	101
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	Mavrična dominacija v leksikografskem produktu grafov
Secondary abstract:	A ▫$k$▫-rainbow dominating function of a graph ▫$G$▫ is a map ▫$f$▫ from ▫$V(G)$▫ to the set of all subsets of ▫$\{1,2,\dots,k\}$▫ such that ▫$\{1,\dots,k\} = \bigcup_{u \in N(v)}f(u)$▫ whenever ▫$v$▫ is a vertex with ▫$f(v) = \emptyset$▫. The ▫$k$▫-rainbow domination number of ▫$G$▫ is the invariant ▫$\gamma_{rk}(G)$▫, which is the minimum sum (over all the vertices of ▫$G$▫) of the cardinalities of the subsets assigned by a ▫$k$▫-rainbow dominating function. We focus on the 2-rainbow domination number of the lexicographic product of graphs and prove sharp lower and upper bounds for this number. In fact, we prove the exact value of ▫$\gamma_{r2}(G \circ H)$▫ in terms of domination invariants of ▫$G$▫ except for the case when ▫$\gamma_{r2}(H) = 3$▫ and there exists a minimum 2-rainbow dominating function of ▫$H$▫ such that there is a vertex in ▫$H$▫ with the label ▫$\{1,2\}$▫.
URN:	URN:SI:UM:
Type (COBISS):	Not categorized
Pages:	str. 2133-2141
Volume:	ǂVol. ǂ161
Issue:	ǂiss. ǂ13-14
Chronology:	2013
ID:	1471833