Related papers: Statistics of paths on graphs with two heavy roots
Given a tree and a set ${\cal P}$ of non-trivial simple paths on it, $VPT({\cal P})$ is the VPT graph (i.e. the vertex intersection graph) of the paths ${\cal P}$ of the tree $T$, and $EPT({\cal P})$ is the EPT graph (i.e. the edge…
The treewidth of a graph is an important invariant in structural and algorithmic graph theory. This paper studies the treewidth of line graphs. We show that determining the treewidth of the line graph of a graph $G$ is equivalent to…
We shall study degree-monotone paths in graphs, a problem inspired by the celebrated theorem of Erd{\H{o}}s-Szekeres concerning the longest monotone subsequence of a given sequence of numbers. A path P in a graph G is said to be a degree…
Characterized are all simple undirected graphs $G$ such that any real symmetric matrix that has graph $G$ has no eigenvalues of multiplicity more than 2. All such graphs are partial 2-trees (and this follows from a result for rather general…
Tree-width and path-width are well-known graph parameters. Many NP-hard graph problems allow polynomial-time solutions, when restricted to graphs of bounded tree-width or bounded path-width. In this work, we study the behavior of tree-width…
We study the graphs formed from instances of the stable matching problem by connecting pairs of elements with an edge when there exists a stable matching in which they are matched. Our results include the NP-completeness of recognizing…
The families EPT (resp. EPG) Edge Intersection Graphs of Paths in a tree (resp. in a grid) are well studied graph classes. Recently we introduced the graph classes Edge-Intersecting and Non-Splitting Paths in a Tree ENPT, and in a Grid…
The intersection graph of a collection of trapezoids with corner points lying on two parallel lines is called a trapezoid graph. These graphs and their generalizations were applied in various fields, including modeling channel routing…
A certain class of directed metric graphs is considered. Asymptotics for a number of possible endpoints of a random walk at large times is found.
We consider random temporal graphs, a version of the classical Erd\H{o}s--R\'enyi random graph G(n,p) where additionally, each edge has a distinct random time stamp, and connectivity is constrained to sequences of edges with increasing time…
A path graph is the intersection graph of paths in a tree. A directed path graph is the intersection graph of paths in a directed tree. Even if path graphs and directed path graphs are characterized very similarly, their recognition…
Trotter and Erd\"os found conditions for when a directed $m \times n$ grid graph on a torus is Hamiltonian. We consider the analogous graphs on a two-holed torus, and study their Hamiltonicity. We find an $\mathcal{O}(n^4)$ algorithm to…
Analysing and explaining relationships between entities in a graph is a fundamental problem associated with many practical applications. For example, a graph of biological pathways can be used for discovering a previously unknown…
Which graphs, in the class of all graphs with given numbers n and m of edges and vertices respectively, minimizes or maximizes the value of some graph parameter? In this paper we develop a technique which provides answers for several…
Huynh, Joret, Micek, Seweryn, and Wollan (Combinatorica, 2022) introduced a graph parameter, later referred to as 2-treedepth and denoted $\mathrm{td}_2(\cdot)$. The parameter is the natural 2-connected version of treedepth. For every…
We consider the problem of determining the inducibility (maximum possible asymptotic density of induced copies) of oriented graphs on four vertices. We provide exact values for more than half of the graphs, and very close lower and upper…
Deciding whether a given graph has a square root is a classical problem that has been studied extensively both from graph theoretic and from algorithmic perspectives. The problem is NP-complete in general, and consequently substantial…
Let $G$ be a graph on $n$ vertices. An induced subgraph $H$ of $G$ is called heavy if there exist two nonadjacent vertices in $H$ with degree sum at least $n$ in $G$. We say that $G$ is $H$-heavy if every induced subgraph of $G$ isomorphic…
We study the effect of the choice of embedding geometry on the entropy of random geometric graph ensembles with soft connection functions. First we show that when the connection range is small, the entropy is dependent only on the dimension…
A degree monotone path in a graph $G$ is a path $P$ such that the sequence of degrees of the vertices in the order in which they appear on $P$ is monotonic. The length of the longest degree monotone path in $G$ is denoted by $mp(G)$. This…