Browsing by Author "Michalski, Adrian"

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Access status: Open Access ,
On minimum intersections of certain secondary dominating sets in graphs
(Wydawnictwa AGH, 2023) Kosiorowska, Anna; Michalski, Adrian; Włoch, Iwona
In this paper we consider secondary dominating sets, also named as $(1,k)$-dominating sets, introduced by Hedetniemi et al. in 2008. In particular, we study intersections of the $(1,1)$-dominating sets and proper $(1,2)$-dominating sets. We introduce $(1,\overline{2})$-intersection index as the minimum possible cardinality of such intersection and determine its value for some classes of graphs.
Access status: Open Access ,
On the existence of independent (1,k) -dominating sets for k∈{1,2} in two products of graphs
(Wydawnictwa AGH, 2026) Bednarz, Paweł; Michalski, Adrian; Paja, Natalia
A subset $J$ of vertices is said to be a $(1,k)$-dominating set if every vertex $v$ not belonging to the set $J$ has a neighbour in $J$ and there exists also another vertex in $J$ within the distance at most $k$ from $v$. In this paper, we study the problem of the existence of independent $(1,k)$-dominating sets for $k\in\{1,2\}$ in the tensor product and in the strong product of two graphs. We give complete characterisations of these graph products, which have independent $(1,1)$-dominating sets or independent $(1,2)$-dominating sets, with respect to the properties of their factors.