TY  - JOUR
ID  - Bednarz2026
LB  - Bednarz2026
AU  - Bednarz, Paweł
AU  - Michalski, Adrian
AU  - Paja, Natalia
TI  - On the existence of independent \((1,k)\)-dominating sets for \(k\in\{1,2\}\) in two products of graphs
ST  - On the existence of independent \((1,k)\)-dominating sets for \(k\in\{1,2\}\) in two products of graphs
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2026
VL  - 46
IS  - 2
SP  - 139
EP  - 152
DO  - https://doi.org/10.7494/OpMath.202601201
UR  - https://doi.org/10.7494/OpMath.202601201
AB  - 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.
KW  - dominating set
KW  - independent set
KW  - multiple domination
KW  - secondary domination
KW  - tensor product
KW  - strong product
SN  - 1232-9274
ER  -