Mutual-visibility in strong products of graphs via total mutual-visibility

Serafino Cicerone (Author), Gabriele Di Stefano (Author), Sandi Klavžar (Author), Ismael G. Yero (Author)

Abstract

Naj bo ▫$G$▫ graf in ▫$X\subseteq V(G)$▫. Potem je ▫$X$▫ množica vzajemne vidnosti, če je vsak par vozlišč iz ▫$X$▫ povezan z najkrajšo potjo brez notranjega vozlišča iz ▫$X$▫. Število vzajemne vidnosti ▫$\mu(G)$▫ v ▫$G$▫ je kardinalnost največje množice vzajemne vidnosti. V tem članku je raziskano število vzajemne vidnosti krepkih produktov grafov. Kot orodje za to so uvedene množice celotne vzajemne vidnosti. Ob tem so predstavljene osnovne lastnosti takšnih množic. Število (skupne) vzajemne vidnosti krepkih produktov je omejeno od spodaj na dva načina in natančno določeno za krepke rešetke poljubne dimenzije. Krepke prizme so obravnavane ločeno in podanih je nekaj strogih mej za njihovo število vzajemne vidnosti.

Keywords

množica vzajemne vidnosti;število vzajemne vidnosti;množica celotne vzajemne vidnosti;krepki produkt grafov;mutual-visibility set;mutual-visibility number;total mutual-visibility set;strong product of graphs;

Data

Language:	English
Year of publishing:	2024
Typology:	1.01 - Original Scientific Article
Organization:	UL FMF - Faculty of Mathematics and Physics
UDC:	519.17
COBISS:	201822467
ISSN:	0166-218X
Views:	32
Downloads:	16
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	Slovenian
Secondary title:	Vzajemna vidnost v krepkih produktih grafov skozi celotno medsebojno vidnost
Secondary abstract:	Let ▫$G$▫ be a graph and ▫$X\subseteq V(G)$▫. Then ▫$X$▫ is a mutual-visibility set if each pair of vertices from ▫$X$▫ is connected by a geodesic with no internal vertex in ▫$X$▫. The mutual-visibility number ▫$\mu(G)$▫ of ▫$G$▫ is the cardinality of a largest mutual-visibility set. In this paper, the mutual-visibility number of strong product graphs is investigated. As a tool for this, total mutual-visibility sets are introduced. Along the way, basic properties of such sets are presented. The (total) mutual-visibility number of strong products is bounded from below in two ways, and determined exactly for strong grids of arbitrary dimension. Strong prisms are studied separately and a couple of tight bounds for their mutual-visibility number are given.
Secondary keywords:	množica vzajemne vidnosti;število vzajemne vidnosti;množica celotne vzajemne vidnosti;krepki produkt grafov;
Type (COBISS):	Article
Pages:	str. 136-146
Issue:	ǂVol. ǂ358
Chronology:	Dec. 2024
DOI:	10.1016/j.dam.2024.06.038
ID:	25225842