# The Dynamics of the Forest Graph Operator

• Volume: 36, Issue: 4, page 899-913
• ISSN: 2083-5892

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## Abstract

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In 1966, Cummins introduced the “tree graph”: the tree graph 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 T1 and T2 are adjacent if T2 = T1 − e + f for some edges e ∈ T1 and f ∉ T1. 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 α, finite or infinite, and let N(G) be the set of all labeled maximal forests of G. The forest graph of G, denoted by F(G), is the graph with vertex set N(G) in which two maximal forests F1, F2 of G form an edge if and only if they differ exactly by one edge, i.e., F2 = F1 − e + f for some edges e ∈ F1 and f ∉ F1. Using the theory of cardinal numbers, Zorn’s lemma, transfinite induction, the axiom of choice and the well-ordering principle, we determine the F-convergence, F-divergence, F-depth and F-stability of any graph G. In particular it is shown that a graph G (finite or infinite) is F-convergent if and only if G has at most one cycle of length 3. The F-stable graphs are precisely K3 and K1. The F-depth of any graph G different from K3 and K1 is finite. We also determine various parameters of F(G) for an infinite graph G, including the number, order, size, and degree of its components.

## How to cite

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Suresh Dara, et al. "The Dynamics of the Forest Graph Operator." Discussiones Mathematicae Graph Theory 36.4 (2016): 899-913. <http://eudml.org/doc/287136>.

@article{SureshDara2016,
abstract = {In 1966, Cummins introduced the “tree graph”: the tree graph 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 T1 and T2 are adjacent if T2 = T1 − e + f for some edges e ∈ T1 and f ∉ T1. 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 α, finite or infinite, and let N(G) be the set of all labeled maximal forests of G. The forest graph of G, denoted by F(G), is the graph with vertex set N(G) in which two maximal forests F1, F2 of G form an edge if and only if they differ exactly by one edge, i.e., F2 = F1 − e + f for some edges e ∈ F1 and f ∉ F1. Using the theory of cardinal numbers, Zorn’s lemma, transfinite induction, the axiom of choice and the well-ordering principle, we determine the F-convergence, F-divergence, F-depth and F-stability of any graph G. In particular it is shown that a graph G (finite or infinite) is F-convergent if and only if G has at most one cycle of length 3. The F-stable graphs are precisely K3 and K1. The F-depth of any graph G different from K3 and K1 is finite. We also determine various parameters of F(G) for an infinite graph G, including the number, order, size, and degree of its components.},
author = {Suresh Dara, S.M. Hegde, Venkateshwarlu Deva, S.B. Rao, Thomas Zaslavsky},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {forest graph operator; graph dynamics},
language = {eng},
number = {4},
pages = {899-913},
title = {The Dynamics of the Forest Graph Operator},
url = {http://eudml.org/doc/287136},
volume = {36},
year = {2016},
}

TY - JOUR
AU - Suresh Dara
AU - S.M. Hegde
AU - Venkateshwarlu Deva
AU - S.B. Rao
AU - Thomas Zaslavsky
TI - The Dynamics of the Forest Graph Operator
JO - Discussiones Mathematicae Graph Theory
PY - 2016
VL - 36
IS - 4
SP - 899
EP - 913
AB - In 1966, Cummins introduced the “tree graph”: the tree graph 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 T1 and T2 are adjacent if T2 = T1 − e + f for some edges e ∈ T1 and f ∉ T1. 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 α, finite or infinite, and let N(G) be the set of all labeled maximal forests of G. The forest graph of G, denoted by F(G), is the graph with vertex set N(G) in which two maximal forests F1, F2 of G form an edge if and only if they differ exactly by one edge, i.e., F2 = F1 − e + f for some edges e ∈ F1 and f ∉ F1. Using the theory of cardinal numbers, Zorn’s lemma, transfinite induction, the axiom of choice and the well-ordering principle, we determine the F-convergence, F-divergence, F-depth and F-stability of any graph G. In particular it is shown that a graph G (finite or infinite) is F-convergent if and only if G has at most one cycle of length 3. The F-stable graphs are precisely K3 and K1. The F-depth of any graph G different from K3 and K1 is finite. We also determine various parameters of F(G) for an infinite graph G, including the number, order, size, and degree of its components.
LA - eng
KW - forest graph operator; graph dynamics
UR - http://eudml.org/doc/287136
ER -

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