Counting independent dominating sets in linear polymers

TL;DR

This paper develops methods to count independent dominating sets in linear and cyclic graph structures, providing explicit formulas and generating functions for specific graph classes.

Contribution

It introduces new recurrence relations and generating functions for counting independent dominating sets in complex graph chains.

Findings

Derived explicit recurrences for various graph classes.

Computed generating functions for triangular and square cacti chains.

Provided enumeration formulas for independent dominating sets in linear and cyclic graphs.

Abstract

A vertex subset $W\subseteq V$ of the graph $G = (V,E)$ is an independent dominating set if every vertex in $V\setminus W$ is adjacent to at least one vertex in $W$ and the vertices of $W$ are pairwise non-adjacent. We enumerate independent dominating sets in several classes of graphs made by a linear or cyclic concatenation of basic building blocks. Explicit recurrences are derived for the number of independent dominating sets of these kind of graphs. Generating functions for the number of independent dominating sets of triangular and squares cacti chain are also computed.