Related papers: Representing Directed Trees as Straight Skeletons
A perfect forest is a spanning forest of a connected graph $G$, all of whose components are induced subgraphs of $G$ and such that all vertices have odd degree in the forest. A perfect forest generalised a perfect matching since, in a…
We introduce topological notions of polytopes and simplexes, the latter being expected to play in p-adically closed fields the role played by real simplexes in the classical results of triangulation of semi-algebraic sets over real closed…
A formally normal weighted shift on a directed tree is shown to be a bounded normal operator. The question of whether a normal extension of a subnormal weighted shift on a directed tree can be modeled as a weighted shift on some, possible…
A monotone drawing of a graph G is a straight-line drawing of G such that, for every pair of vertices u,w in G, there exists abpath P_{uw} in G that is monotone in some direction l_{uw}. (Namely, the order of the orthogonal projections of…
We analyze a class of spatial random spanning trees built on a realization of a homogeneous Poisson point process of the plane. This tree has a simple radial structure with the origin as its root. We first use stochastic geometry arguments…
When representing a solid object there are alternatives to the use of traditional explicit (surface meshes) or implicit (zero crossing of implicit functions) methods. Skeletal representations encode shape information in a mixed fashion:…
Rectangulations are decompositions of a square into finitely many axis-aligned rectangles. We describe realizations of $(n-1)$-dimensional polytopes associated with two combinatorial families of rectangulations composed of $n$ rectangles.…
Let $C_{k_1}, \ldots, C_{k_n}$ be cycles with $k_i\geq 2$ vertices ($1\le i\le n$). By attaching these $n$ cycles together in a linear order, we obtain a graph called a polygon chain. By attaching these $n$ cycles together in a cyclic…
Reconfiguration problems involve determining whether two given configurations can be transformed into each other under specific rules. The Token Sliding problem asks whether, given two different set of tokens on vertices of a graph $G$, we…
We denote a polygon as a connected component in Cayley tree of order 2 containing certain number of fix vertices. We found an exact formula for a polygon counting problem for two cases, in which, for the first case the polygon contain a…
Given a tree $T$, its path polytope is the convex hull of the edge indicator vectors for the paths between any two distinct leaves in $T$. These polytopes arise naturally in polyhedral geometry and applications, such as phylogenetics,…
\textsc{Directed Token Sliding} asks, given a directed graph and two sets of pairwise nonadjacent vertices, whether one can reach from one set to the other by repeatedly applying a local operation that exchanges a vertex in the current set…
We study the problem of learning a node-labeled tree given independent traces from an appropriately defined deletion channel. This problem, tree trace reconstruction, generalizes string trace reconstruction, which corresponds to the tree…
Let $G$ be a group. The directed endomorphism graph, $\dend(G)$ of $G$ is a directed graph with vertex set $G$ and there is a directed edge from the vertex $a$ to the vertex $b$ if $a \neq b$ and there exists an endomorphism on $G$ mapping…
A tree with at most k leaves is called k-ended tree, and a tree with exactly k leaves is called k-end tree, where a leaf is a vertex of degree one. Contraction of a graph G along the edge e means deleting the edge e and identifying its end…
This paper outlines a method to determine whether two label-regular directed trees, are isomorphic and when they are almost isomorphic. The approach involves reinterpreting label-regular directed trees as universal covers of rooted graphs.…
Given a directed graph, G=(V,E), a path query, path(u,v), returns whether there is a directed path from u to v in G, for u,v vertices in V. Given only V, exactly learning all the edges in G using path queries is often impossible, since path…
The tree reconstruction problem is to find an embedded straight-line tree that approximates a given cloud of unorganized points in $\mathbb{R}^m$ up to a certain error. A practical solution to this problem will accelerate a discovery of new…
Deciding whether a family of disjoint line segments in the plane can be linked into a simple polygon (or a simple polygonal chain) by adding segments between their endpoints is NP-hard.
It is well known that directed treewidth does not enjoy the nice algorithmic properties of its undirected counterpart. There exist, however, some positive results that, essentially, present XP algorithms for the problem of finding, in a…