# The growth rate over trees of any family of set defined by a monadic second order formula is semi-computable

Monadic second order logic can be used to express many classical notions of sets of vertices of a graph as for instance: dominating sets, induced matchings, perfect codes, independent sets or irredundant sets. Bounds on the number of sets of any such family of sets are interesting from a combinatorial point of view and have algorithmic applications. Many such bounds on different families of sets over different classes of graphs are already provided in the literature. In particular, Rote recently showed that the number of minimal dominating sets in trees of order n is at most 95^n/13 and that this bound is asymptotically sharp up to a multiplicative constant. We build on his work to show that what he did for minimal dominating sets can be done for any family of sets definable by a monadic second order formula. We first show that, for any monadic second order formula over graphs that characterizes a given kind of subset of its vertices, the maximal number of such sets in a tree can be expressed as the growth rate of a bilinear system. This mostly relies on well known links between monadic second order logic over trees and tree automata and basic tree automata manipulations. Then we show that this “growth rate” of a bilinear system can be approximated from above. We then use our implementation of this result to provide bounds (some sharp and some almost sharp) on the number of independent dominating sets, total perfect dominating sets, induced matchings, maximal induced matchings, minimal perfect dominating sets, perfect codes and maximal irredundant sets on trees. We also solve a question from D. Y. Kang et al. regarding r-matchings. Remark that this approach is easily generalizable to graphs of bounded tree width or clique width (or any similar class of graphs where tree automata are meaningful).

## Authors

• 9 publications
03/11/2019

### Minimal Dominating Sets in a Tree: Counting, Enumeration, and Extremal Results

A tree with n vertices has at most 95^n/13 minimal dominating sets. The ...
05/07/2018

### Domination Cover Number of Graphs

A set D ⊆ V for the graph G=(V, E) is called a dominating set if any ver...
05/07/2018

### Neighborhood preferences for minimal dominating sets enumeration

We investigate two different approaches to enumerate minimal dominating ...
11/11/2021

### A Note on the Maximum Number of Minimal Connected Dominating Sets in a Graph

We prove constructively that the maximum possible number of minimal conn...
04/29/2022

### Enumerating Connected Dominating Sets

The question to enumerate all inclusion-minimal connected dominating set...
06/25/2019

### Asymptotic growth rate of square grids dominating sets: a symbolic dynamics approach

In this text, we prove the existence of an asymptotic growth rate of the...
08/29/2018

### Asymmetry of copulas arising from shock models

When choosing the right copula for our data a key point is to distinguis...
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