The k-Power Domination Number in Some Self-Similar Graphs

TL;DR

This paper calculates the exact $k$-power domination number in specific self-similar graphs, revealing different behaviors in scale-free and fractal structures, with implications for graph theory applications.

Contribution

It provides the first exact calculations of the $k$-power domination number in pseudofractal scale-free web and Sierpiński gasket graphs, highlighting their differing properties.

Findings

$k$-power domination number is 1 for $k extgreater=2$ in Sierpiński gasket.

In pseudofractal scale-free web, the number grows exponentially with vertices.

Scale-free property influences the $k$-power domination behavior.

Abstract

The $k$-power domination problem is a problem in graph theory, which has applications in many areas. However, it is hard to calculate the exact $k$-power domination number since determining k-power domination number of a generic graph is a NP-complete problem. We determine the exact $k$-power domination number in two graphs which have the same number of vertices and edges: pseudofractal scale-free web and Sierpi\'nski gasket. The $k$-power domination number becomes 1 for $k\ge2$ in the Sierpi\'nski gasket, while the $k$-power domination number increases at an exponential rate with regard to the number of vertices in the pseudofractal scale-free web. The scale-free property may account for the difference in the behavior of two graphs.