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Query: "keywords" (metric dimension) .

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1.
On metric dimensions of hypercubes
Aleksander Kelenc, Aoden Teo Masa Toshi, Riste Škrekovski, Ismael G. Yero, 2023, original scientific article

Abstract: In this note we show two unexpected results concerning the metric, the edge metric and the mixed metric dimensions of hypercube graphs. First, we show that the metric and the edge metric dimensions of $Q_d$ differ by at most one for every integer $d$. In particular, if $d$ is odd, then the metric and the edge metric dimensions of $Q_d$ are equal. Second, we prove that the metric and the mixed metric dimensions of the hypercube $Q_d$ are equal for every $d \ge 3$. We conclude the paper by conjecturing that all these three types of metric dimensions of $Q_d$ are equal when d is large enough.
Keywords: edge metric dimension, mixed metric dimension, metric dimension, hypercubes
Published in DiRROS: 19.03.2024; Views: 113; Downloads: 34
.pdf Full text (259,98 KB)

2.
Resolvability and convexity properties in the Sierpiński product of graphs
Michael A. Henning, Sandi Klavžar, Ismael G. Yero, 2024, original scientific article

Abstract: Let $G$ and $H$ be graphs and let $f \colon V(G)\rightarrow V(H)$ be a function. The Sierpiński product of $G$ and $H$ with respect to $f$, denoted by $G \otimes _f H$, is defined as the graph on the vertex set $V(G)\times V(H)$, consisting of $|V(G)|$ copies of $H$; for every edge $gg'$ of $G$ there is an edge between copies $gH$ and $g'H$ of $H$ associated with the vertices $g$ and $g'$ of $G$, respectively, of the form $(g,f(g'))(g',f(g))$. 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.
Keywords: Sierpiński product of graphs, metric dimension, trees, convex subgraph
Published in DiRROS: 16.02.2024; Views: 135; Downloads: 58
.pdf Full text (432,07 KB)
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