Related papers: A note on the flip distance between non-crossing s…
For a set $P$ of $n$ points in the plane in general position, a non-crossing spanning tree is a spanning tree of the points where every edge is a straight-line segment between a pair of points and no two edges intersect except at a common…
A non-crossing spanning tree of a set of points in the plane is a spanning tree whose edges pairwise do not cross. Avis and Fukuda in 1996 proved that there always exists a flip sequence of length at most $2n-4$ between any pair of…
For a set $P$ of $n$ points in general position in the plane, the flip graph $F(P)$ has a vertex for each non-crossing spanning tree on $P$ and an edge between any two spanning trees that can be transformed into each other by one edge flip.…
A flip in a plane spanning tree $T$ is the operation of removing one edge from $T$ and adding another edge such that the resulting structure is again a plane spanning tree. For trees on a set of points in convex position we study two…
We consider the problem of reconfiguring non-crossing spanning trees on point sets. For a set $P$ of $n$ points in general position in the plane, the flip graph $F(P)$ has a vertex for each non-crossing spanning tree on $P$ and an edge…
Flips in triangulations of convex polygons arise in many different settings. They are isomorphic to rotations in binary trees, define edges in the 1-skeleton of the Associahedron and cover relations in the Tamari Lattice. The complexity of…
This paper is a contribution to the problem of counting geometric graphs on point sets. More concretely, we look at the maximum numbers of non-crossing spanning trees and forests. We show that the so-called double chain point configuration…
A spanning tree of an unweighted graph is a minimum average stretch spanning tree if it minimizes the ratio of sum of the distances in the tree between the end vertices of the graph edges and the number of graph edges. We consider the…
We investigate the complexity of finding a transformation from a given spanning tree in a graph to another given spanning tree in the same graph via a sequence of edge flips. The exchange property of the matroid bases immediately yields…
We study an old geometric optimization problem in the plane. Given a perfect matching $M$ on a set of $n$ points in the plane, we can transform it to a non-crossing perfect matching by a finite sequence of flip operations. The flip…
We obtain new lower and upper bounds for the maximum multiplicity of some weighted and, respectively, non-weighted common geometric graphs drawn on n points in the plane in general position (with no three points collinear): perfect…
We develop combinatorial methods for computing the rotation distance between binary trees, i.e., equivalently, the flip distance between triangulations of a polygon. As an application, we prove that, for each n, there exist size n trees at…
Codes over trees were introduced recently to bridge graph theory and coding theory with diverse applications in computer science and beyond. A central challenge lies in determining the maximum number of labelled trees over $n$ nodes with…
We study the reconfiguration of plane spanning trees on point sets in the plane in convex position, where a reconfiguration step (flip) replaces one edge with another, yielding again a plane spanning tree. The flip distance between two…
We study spanners in planar domains, including polygonal domains, polyhedral terrain, and planar metrics. Previous work showed that for any constant $\epsilon\in (0,1)$, one could construct a $(2+\epsilon)$-spanner with $O(n\log(n))$ edges…
A spanning tree of a graph $G$ is a connected acyclic spanning subgraph of $G$. We consider enumeration of spanning trees when $G$ is a $2$-tree, meaning that $G$ is obtained from one edge by iteratively adding a vertex whose neighborhood…
Three standard subtree transfer operations for binary trees, used in particular for phylogenetic trees, are: tree bisection and reconnection ($TBR$), subtree prune and regraft ($SPR$) and rooted subtree prune and regraft ($rSPR$). For a…
The operation of transforming one spanning tree into another by replacing an edge has been considered widely, both for general and planar straight-line graphs. For the latter, several variants have been studied (e.g., edge slides and edge…
As well known the rotation distance D(S,T) between two binary trees S, T of n vertices is the minimum number of rotations of pairs of vertices to transform S into T. We introduce the new operation of chain rotation on a tree, involving two…
Let $G$ be a connected graph with vertex set $V(G)$, and denote by $d_G(u,v)$ the distance from $u$ to $v$ in $G$, for any $u,v \in V(G)$. The average distance of an $n$-vertex connected graph $G$, denoted by $\mu(G)$, is defined to be the…