Iwona Włoch


Title: On \((1,2)\)-dominating sets and their variants in graphs

Abstract:

The concept of a \((1,2)\)-dominating sets was introduced in [3]. Let \(k\) be an integer. A subset \(D \subseteq V(G)\) is a \((1,k)\)-dominating set of a graph \(G\) if for each vertex \(v \in V(G)\setminus D\) thete is \(u,w \in D\) such that \(vu \in E(G)\) and \(d_G(v,w) \leq k\). If \(k=1\), then we obtain the definition of \(2\)-dominating set of \(G\). We present some results concerning \((1,2)\)-dominating sets and their variants: proper \((1,2)\)-dominating sets and strong \((1,2)\)-dominating sets.

References:
  1. [1] U. Bednarz, I. W loch, On strong (1,1,2)-kernels in graphs, Ars Combinatoria, 152, 32-43, 2020.
  2. [2] U. Bednarz, I. W loch, Fibonacci numbers in graphs with strong (1,1,2)-kernels, Bol. Soc. Mat. Mex. 27, 12 (2021).
  3. [3] T. W. Haynes, S. T. Hedetniemi, D. Knisely, D. F. Rall, Secondary domination in graphs, AKCE International Journal of Graphs and Combinatorics 5 (2008), 103-115
  4. [4] A. Michalski, I. Wloch, On the existence and the number of independent (1,2)- dominating sets in the G-join of graphs, Applied Mathematics and Computation 377 (2020) 125155.
  5. [5] A. Michalski, P. Bednarz, On independent secondary dominating sets in generalized graph products, Symmetry, 2021, 13(12), 2399.
  6. [6] A. Michalski, I. W loch, M. Dettlaff, M. Lema´nska, On proper (1,2)-dominating sets, Mathematical Methods in the Applied Sciences, 2022;45(11):7050-7057.

Iwona Włoch is an associate Professor at the Department of Discrete Mathematics, Faculty of Mathematics and Applied Physics, Rzeszow University of Technlogy, Poland. Ph. D. in 1998 at Wroclaw University of Science and Technology, Poland. Habilitation in Mathematics (2011); UPJS Kosice, Slovakia. Main research topics include domination and independence in graphs, Fibonacci type sequences and hypercomplex numbers. The author or co-author over 60 publications, advisor of two doctoral dissertations.
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