Related papers: Intersection representation of digraphs in trees w…
For stacked simplicial complexes, (special subclasses of such are: trees, triangulations of polygons, stacked polytopes), we give an explicit bijection between partitions of facets (for trees: edges), and partitions of vertices into…
An outerstring graph is the intersection graph of curves lying inside a disk with one endpoint on the boundary of the disk. We show that an outerstring graph with $n$ vertices has treewidth $O(\alpha\log n)$, where $\alpha$ denotes the…
This paper addresses the following questions for a given tree $T$ and integer $d\geq2$: (1) What is the minimum number of degree-$d$ subtrees that partition $E(T)$? (2) What is the minimum number of degree-$d$ subtrees that cover $E(T)$? We…
A linear forest is a collection of vertex-disjoint paths. The Linear Arboricity Conjecture states that every graph of maximum degree $\Delta$ can be decomposed into at most $\lceil(\Delta+1)/2\rceil$ linear forests. We prove that $\Delta/2…
The number of topologically different plane real algebraic curves of a given degree $d$ has the form $\exp(C d^2 + o(d^2))$. We determine the best available upper bound for the constant $C$. This bound follows from Arnold inequalities on…
Let T be a (not necessarily positive) weighted tree with n leaves numbered by the set {1,...,n}. Define the k-weights of the tree D_{i_1,....,i_k}(T) as the sum of the lengths of the edges of the minimal subtree connecting i_1,....,i_k. We…
Color-constrained subgraph problems are those where we are given an edge-colored (directed or undirected) graph and the task is to find a specific type of subgraph, like a spanning tree, an arborescence, a single-source shortest path tree,…
We show that for any Riemannian foliation with a simply connected and negatively curved leaf space the normal exponential map of a leaf is a diffeomorphism. As an application, if the leaves are furthermore minimal submanifolds, we give a…
The L-intersection graphs are the graphs that have a representation as intersection graphs of axis parallel shapes in the plane. A subfamily of these graphs are {L, |, --}-contact graphs which are the contact graphs of axis parallel L, |,…
Given an ordered partition $\Pi =\{P_1,P_2, ...,P_t\}$ of the vertex set $V$ of a connected graph $G=(V,E)$, the \emph{partition representation} of a vertex $v\in V$ with respect to the partition $\Pi$ is the vector…
In this paper we characterize the unique graph whose algebraic connectivity is minimum among all connected graphs with given order and fixed matching number or edge covering number, and present two lower bounds for the algebraic…
For a weighted digraph without loops $V$, the arc weights of which can be obtained from an undirected graph with loops ${\sf P}$ according to the rule $v_{ij}=p_{ij}-p_{ii}$, the properties are studied. An effective algorithm for…
Let $T$ be a tree with $t$ edges. We show that the number of isomorphic (labeled) copies of $T$ in a graph $G = (V,E)$ of minimum degree at least $t$ is at least \[2|E| \prod_{v \in V} (d(v) - t + 1)^{\frac{(t-1)d(v)}{2|E|}}.\]…
An L-shape is the union of a horizontal and a vertical segment with a common endpoint. These come in four rotations: L, \Gamma, LE{} and \eeG. A $k$-bend path is a simple path in the plane, whose direction changes $k$ times from horizontal…
Given a set of nonempty subsets of some universal set, their intersection graph is defined as the graph with one vertex for each set and two vertices are adjacent precisely when their representing sets have non-empty intersection. Sometimes…
In this paper, we investigate the computational complexity of subgraph reconfiguration problems in directed graphs. More specifically, we focus on the problem of reconfiguring arborescences in a digraph, where an arborescence is a directed…
Understanding the evolution of a set of genes or species is a fundamental problem in evolutionary biology. The problem we study here takes as input a set of trees describing {possibly discordant} evolutionary scenarios for a given set of…
We prove that, if $m$ is sufficiently large, every graph on $m+1$ vertices that has a universal vertex and minimum degree at least $\lfloor \frac{2m}{3} \rfloor$ contains each tree $T$ with $m$ edges as a subgraph. Our result confirms, for…
In this paper, we present a complete characterization of mutual-visibility sets in trees. It is shown that a subset $S$ is a mutual-visibility set of a tree $T$ if and only if it coincides with the set of leaves of the Steiner subtree…
Path graphs are intersection graphs of paths in a tree. We start from the characterization of path graphs by Monma and Wei [C.L.~Monma,~and~V.K.~Wei, Intersection Graphs of Paths in a Tree, J. Combin. Theory Ser. B, 41:2 (1986) 141--181]…