The Dynamics of the Forest Graph Operator

• Autorzy: Suresh Dara , S.M. Hegde , Venkateshwarlu Deva , S.B. Rao , Thomas Zaslavsky
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

forest graph operator; graph dynamics

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 2016
• Tom: 36
• Numer: 4
• Strony: 899-913

Daty

wydano

2016-11-01

otrzymano

2015-07-31

poprawiono

2016-01-06

zaakceptowano

2016-01-06

online

2016-10-21

Twórcy

autor

Suresh Dara

• Department of Mathematical and Computational Sciences National Institute of Technology Karnataka Surathkal, Mangalore-575025,

autor

S.M. Hegde

• Department of Mathematical and Computational Sciences National Institute of Technology Karnataka Surathkal, Mangalore-575025,

autor

Venkateshwarlu Deva

• C.R. Rao Advanced Institute of Mathematics, Statistics and Computer Science Hyderabad-500 046,

autor

S.B. Rao

• C. R. Rao Advanced Institute of Mathematics, Statistics and Computer Science Hyderabad-500 046,

autor

Thomas Zaslavsky

• Department of Mathematical Sciences, Binghamton University Binghamton, NY, 13902-6000,

Identyfikatory

DOI

10.7151/dmgt.1901