Related papers: Long induced paths in a configuration model
We prove that every 3-connected planar graph on $n$ vertices contains an induced path on $\Omega(\log n)$ vertices, which is best possible and improves the best known lower bound by a multiplicative factor of $\log \log n$. We deduce that…
Paths are important structural elements in complex networks because they are finite (unlike walks), related to effective node coverage (minimum spanning trees), and can be understood as being dual to star connectivity. This article…
An extension of an induced path $P$ in a graph $G$ is an induced path $P'$ such that deleting the endpoints of $P'$ results in $P$. An induced path in a graph is said to be avoidable if each of its extensions is contained in an induced…
We study {\em routing} and {\em scheduling} in packet-switched networks. We assume an adversary that controls the injection time, source, and destination for each packet injected. A set of paths for these packets is {\em admissible} if no…
The Longest Path Problem is a question of finding the maximum length between pairs of vertices of a graph. In the general case, the problem is NP-complete. However, there is a small collection of graph classes for which there exists an…
More than 40 years ago, Galvin, Rival and Sands showed that every $K_{s, s}$-free graph containing an $n$-vertex path must contain an induced path of length $f(n)$, where $f(n)\to \infty$ as $n\to \infty$. Recently, it was shown by Duron,…
For a graph $G$ and $p\in [0,1]$, let $G_p$ arise from $G$ by deleting every edge mutually independently with probability $1-p$. The random graph model $(K_n)_p$ is certainly the most investigated random graph model and also known as the…
Diameter -- the task of computing the length of a longest shortest path -- is a fundamental graph problem. Assuming the Strong Exponential Time Hypothesis, there is no $O(n^{1.99})$-time algorithm even in sparse graphs [Roditty and…
Motivated by an old question of Gallai (1966) on the intersection of longest paths in a graph and the well-known conjectures of Lov\'{a}sz (1969) and Thomassen (1978) on the maximum length of paths and cycles in vertex-transitive graphs, we…
We discuss several classical results about long paths and Hamilton cycles in random graphs and present accessible versions of their proofs, relying on the Depth First Search (DFS) algorithm and the notion of boosters.
The age $\mathcal{A}(G)$ of a graph $G$ (undirected and without loops) is the collection of finite induced subgraphs of $G$, considered up to isomorphy and ordered by embeddability. It is well-quasi-ordered (wqo) for this order if it…
Given an undirected graph $G=(V,E)$, the longest induced path problem (LIPP) consists of obtaining a maximum cardinality subset $W\subseteq V$ such that $W$ induces a simple path in $G$. In this paper, we propose two new formulations with…
The Price model, the directed version of the Barab\'{a}si-Albert model, produces a growing directed acyclic graph. We look at variants of the model in which directed edges are added to the new vertex in one of two ways: using cumulative…
We present progress on three old conjectures about longest paths and cycles in graphs. The first pair of conjectures, due to Lov\'{a}sz from 1969 and Thomassen from 1978, respectively, states that all connected vertex-transitive graphs…
For the task of moving a group of indistinguishable agents on a connected graph with unit edge lengths into an arbitrary goal formation, it was previously shown that distance optimal paths can be scheduled to complete with a tight…
The goal of this paper is to design a simplex algorithm for linear programs on lattice polytopes that traces `short' simplex paths from any given vertex to an optimal one. We consider a lattice polytope $P$ contained in $[0,k]^n$ and…
We introduce an algorithm that constructs a random uniform graph with prescribed degree sequence together with a depth first exploration of it. In the so-called supercritical regime where the graph contains a giant component, we prove that…
We devise an algorithm that approximately computes the number of paths of length $k$ in a given directed graph with $n$ vertices up to a multiplicative error of $1 \pm \varepsilon$. Our algorithm runs in time $\varepsilon^{-2} 4^k(n+m)…
Prime path coverage is a powerful structural testing criterion, but generating all prime paths in a directed graph remains computationally challenging due to the potentially exponential number of them. Existing approaches typically rely on…
It is easy to see that in a connected graph any 2 longest paths have a vertex in common. For k>=7, Skupien in [7] obtained a connected graph in which some k longest paths have no common vertex, but every k-1 longest paths have a common…