Author:
Henning Michael A.,Klavžar Sandi,Yero Ismael G.
Abstract
AbstractLet G and H be graphs and let $$f :V(G)\rightarrow V(H)$$
f
:
V
(
G
)
→
V
(
H
)
be a function. The Sierpiński product of G and H with respect to f, denoted by $$G \otimes _f H$$
G
⊗
f
H
, is defined as the graph on the vertex set $$V(G)\times V(H)$$
V
(
G
)
×
V
(
H
)
, consisting of |V(G)| copies of H; for every edge $$gg'$$
g
g
′
of G there is an edge between copies gH and $$g'H$$
g
′
H
of H associated with the vertices g and $$g'$$
g
′
of G, respectively, of the form $$(g,f(g'))(g',f(g))$$
(
g
,
f
(
g
′
)
)
(
g
′
,
f
(
g
)
)
. The Sierpiński metric dimension and the upper Sierpiński metric dimension of two graphs are determined. Closed formulas are determined for Sierpiński products of trees, and for Sierpiński products of two cycles where the second factor is a triangle. We also prove that the layers with respect to the second factor in a Sierpiński product graph are convex.
Funder
Javna Agencija za Raziskovalno Dejavnost RS
Ministerio de Ciencia e Innovación
Publisher
Springer Science and Business Media LLC