On the Fixed Degree Tree Graph

Abstract

A 2-switch on a simple graph G consists of deleting two edges {u, v} and {x, y} of G and adding the edges {u, x} and {v, y}, provided the resulting graph is a simple graph. It is well known that if two graphs G and H have the same set of vertices and the same degree sequence, then H can be obtained from G by a finite sequence of 2-switches. While the 2-switch transformation preserves the degree sequence other conditions like connectivity may be lost. We study the restricted case where 2-switches are applied to trees to obtain trees.

The biplanar tree graph

“…Given a connected graph G the (combinatorial) tree graph T ðGÞ is the graph whose vertices are the spanning trees of G and two trees P and Q are adjacent in T ðGÞ if there are edges e 2 P and f 2 Q such that Q ¼ P À e þ f . This graph has been widely studied as well as some of its variations, like the adjacency tree graph [10,16], the leaf exchange tree graph [2,7], the tree graph defined by a set of cycles [5,12] and the fixed degree tree graph [6], among others. Avis and Fukuda [1] studied the geometric version of the combinatorial problem: given a set P of points in general position, the graph T ðPÞ is the subgraph of T ðnÞ induced by the plane trees.…”

Graphs Isomorphisms Under Edge-Replacements and the Family of Amoebas

“…We call such edge substitutions feasible edge-replacements. Similar edge-operations have been studied, for instance, in [13,18,19,28,37,39]. As introduced in [11], a family F of graphs, all of them having the same number of edges, is called closed in a graph H if, for every two copies F, F of members of F contained in H, there is a chain of graphs H 1 , H 2 , .…”