The Dynamics of the Forest Graph Operator

TL;DR

This paper introduces the forest graph operator, analyzing the convergence, divergence, stability, and depth of graphs based on their maximal forests, revealing specific structural properties and parameters for finite and infinite graphs.

Contribution

It extends the concept of the tree graph to the forest graph, providing a comprehensive set of results on its properties using advanced set theory and graph theory techniques.

Findings

A graph is F-convergent iff it has at most one cycle of length 3.

F-stable graphs are exactly K_3 and K_1.

The F-depth of any graph other than K_3 and K_1 is finite.

Abstract

In 1966, Cummins introduced the "tree graph": the tree graph $\mathbf{T}(G)$ of a graph $G$ (possibly infinite) has all its spanning trees as vertices, and distinct such trees correspond to adjacent vertices if they differ in just one edge, i.e., two spanning trees $T_1$ and $T_2$ are adjacent if $T_2 = T_1 -e +f$ for some edges $e\in T_1$ and $f\notin T_1$. The tree graph of a connected graph need not be connected. To obviate this difficulty we define the "forest graph": let $G$ be a labeled graph of order $\alpha$, finite or infinite, and let $\mathfrak{N}(G)$ be the set of all labeled maximal forests of $G$. The forest graph of $G$, denoted by $\mathbf{F}(G)$, is the graph with vertex set $\mathfrak{N}(G)$ in which two maximal forests $F_1$, $F_2$ of $G$ form an edge if and only if they differ exactly by one edge, i.e., $F_2 = F_1 -e +f$ for some edges $e\in F_1$ and $f\notin F_1$. Using the theory of cardinal numbers, Zorn's lemma, transfinite induction, the axiom of choice and the well-ordering principle, we determine the $\mathbf{F}$-convergence, $\mathbf{F}$-divergence, $\mathbf{F}$-depth and $\mathbf{F}$-stability of any graph $G$. In particular it is shown that a graph $G$ (finite or infinite) is $\mathbf{F}$-convergent if and only if $G$ has at most one cycle of length 3. The $\mathbf{F}$-stable graphs are precisely $K_3$ and $K_1$. The $\mathbf{F}$-depth of any graph $G$ different from $K_3$ and $K_1$ is finite. We also determine various parameters of $\mathbf{F}(G)$ for an infinite graph $G$, including the number, order, size, and degree of its components.