Related papers: Cut Locus Realizations on Convex Polyhedra

Property $(P)$, introduced in recent work and rooted in the classical theory of Parter vertices, concerns the existence of a nonsingular matrix $A\in S(G)$ for which every vertex of $G$ is a $P$-vertex. Previous investigations have fully…

A {\it tiered graph} $G=(V,E)$ with $m $ tiers is a simple graph with $V\subseteq \brk{n}$, where $\brk{n}=\{1,2,\cdots,n\}$, and with a surjective map $t$ from $V$ to $\brk{m}$ such that if $v$ is a vertex adjacent to $v'$ in $G$ with…

We introduce a method for creating a special type of tree, called a tree position, from a weighted graph. Leaves of the tree correspond to vertices of the original graph, and the tree edges contain information which can be used to partition…

In this paper we prove that for a general tree $T$, if $A$ is T-TP, all the submatrices of $A$ associated with the deletion of pendant vertices are $P$-matrices, and $\det A>0$, then the smallest eigenvalue has an eigenvector signed…

The relaxation complexity $\mathrm{rc}(X)$ of the set of integer points $X$ contained in a polyhedron is the smallest number of facets of any polyhedron $P$ such that the integer points in $P$ coincide with $X$. It is a useful tool to…

We study the geometry of metrics and convexity structures on the space of phylogenetic trees, which is here realized as the tropical linear space of all \ ultrametrics. The ${\rm CAT}(0)$-metric of Billera-Holmes-Vogtman arises from the…

We consider, for complete bipartite graphs, the convex hulls of characteristic vectors of all matchings, extended by a binary entry indicating whether the matching contains two specific edges. These polytopes are associated to the quadratic…

We consider the problem of determining the length of the shortest paths between points on the surfaces of tetrahedra and cubes. Our approach parallels the concept of Alexandrov's star unfolding but focuses on specific polyhedra and uses…

In this paper, we present a polynomial dynamic programming algorithm that tests whether a $n$-vertex directed tree $T$ has an upward planar embedding into a convex point-set $S$ of size $n$. Further, we extend our approach to the class of…

We characterize all partitions of the complete twisted graph $T_{2n}$ into plane spanning trees. In the case of partitions of $T_{2n}$ into isomorphic plane spanning trees, we show that all trees in these partitions must be balanced double…

Let $P$ be a convex polyhedron in $\mathbb{R}^3$. The skeleton of $P$ is the graph whose vertices and edges are the vertices and edges of $P$, respectively. We prove that, if these vertices are on the unit-sphere, the skeleton is a $(0.999…

We prove a fix point theorem for monoids of self-embeddings of trees. As a corollary, we obtain a result by Laflamme, Pouzet and Sauer that a tree either contains a subdivided binary tree as a subtree or has a vertex, and edge, an end or…

We introduce a natural notion of depth that applies to individual cutting planes as well as entire families. This depth has nice properties that make it easy to work with theoretically, and we argue that it is a good proxy for the practical…

Let $X$ be a connected compact complex manifold admitting a finite surjective map $A \to X$ from a complex torus $A.$ We prove that up to finite \'etale cover, $X$ is a product of projective spaces and a torus.

Two lattice points are visible to one another if there exist no other lattice points on the line segment connecting them. In this paper we study convex lattice polygons that contain a lattice point such that all other lattice points in the…

It is a classical result that an unrooted tree $T$ having positive real-valued edge lengths and no vertices of degree two can be reconstructed from the induced distance between each pair of leaves. Moreover, if each non-leaf vertex of $T$…

Among all torus links, we characterise those arising as links of simple plane curve singularities by the property that their fibre surfaces admit only a finite number of cutting arcs that preserve fibredness. The same property allows a…

We establish a bound of $O(n^2k^{1+\eps})$, for any $\eps>0$, on the combinatorial complexity of the set $\T$ of line transversals of a collection $\P$ of $k$ convex polyhedra in $\reals^3$ with a total of $n$ facets, and present a…

A stacking operation adds a $d$-simplex on top of a facet of a simplicial $d$-polytope while maintaining the convexity of the polytope. A stacked $d$-polytope is a polytope that is obtained from a $d$-simplex and a series of stacking…

The type C_n full root polytope is the convex hull in R^n of the origin and the points e_i-e_j, e_i+e_j, 2e_k for 1 <= i < j <= n, k \in [n]. Given a graph G, with edges labeled positive or negative, associate to each edge e of G a vector…