Related papers: Roots of Two-Terminal Reliability

Let G be a connected, loopless multigraph. The sandpile group of G is a finite abelian group associated to G whose order is equal to the number of spanning trees in G. Holroyd et al. used a dynamical process on graphs called rotor-routing…

We extend a result of Lyons (2016) from fractional tiling of finite graphs to a version for infinite random graphs. The most general result is as follows. Let $\bf P$ be a unimodular probability measure on rooted networks $(G, o)$ with…

For a connected graph, a path containing all vertices is known as \emph{Hamiltonian path}. For general graphs, there is no known necessary and sufficient condition for the existence of Hamiltonian paths and the complexity of finding a…

In the problem Fault-Tolerant Path (FTP), we are given an edge-weighted directed graph G = (V, E), a subset U \subseteq E of vulnerable edges, two vertices s, t \in V, and integers k and \ell. The task is to decide whether there exists a…

We consider the following activation process in undirected graphs: a vertex is active either if it belongs to a set of initially activated vertices or if at some point it has at least $r$ active neighbors. A \emph{contagious set} is a set…

We show that every connected graph can be approximated by a normal tree, up to some arbitrarily small error phrased in terms of neighbourhoods around its ends. The existence of such approximate normal trees has consequences of both…

Given an increasing graph property $\mathcal{P}$, a graph $G$ is $\alpha$-resilient with respect to $\mathcal{P}$ if, for every spanning subgraph $H\subseteq G$ where each vertex keeps more than a $(1-\alpha)$-proportion of its neighbours,…

In this work, we relate girth and path-degeneracy in classes with sub-exponential expansion, with explicit bounds for classes with polynomial expansion and proper minor-closed classes that are tight up to a constant factor (and tight up to…

Let $P$ be a graph with a vertex $v$ such that $P\backslash v$ is a forest, and let $Q$ be an outerplanar graph. We prove that there exists a number $p=p(P,Q)$ such that every 2-connected graph of path-width at least $p$ has a minor…

In this paper, we disprove the long-standing conjecture that any complete geometric graph on $2n$ vertices can be partitioned into $n$ plane spanning trees. Our construction is based on so-called bumpy wheel sets. We fully characterize…

We present an example of a result in graph theory that is used to obtain a result in another branch of mathematics. More precisely, we show that the isomorphism of certain directed graphs implies that some trinomials over finite fields have…

P-time event graphs are discrete event systems able to model cyclic production systems where tasks need to be performed within given time windows. Consistency is the property of admitting an infinite execution of such tasks that does not…

We consider a Gibbs distribution over all spanning trees of an undirected, edge weighted finite graph, where, up to normalization, the probability of each tree is given by the product of its edge weights. Defining the weighted degree of a…

A spanning tree $T$ of a connected graph $G$ is a subgraph of $G$ that is a tree covers all vertices of $G$. The leaf distance of $T$ is defined as the minimum of distances between any two leaves of $T$. A fractional matching of a graph $G$…

An electrical network with the structure of a random tree is considered: starting from a root vertex, in one iteration each leaf (a vertex with zero or one adjacent edges) of the tree is extended by either a single edge with probability $p$…

A covariance graph is an undirected graph associated with a multivariate probability distribution of a given random vector where each vertex represents each of the different components of the random vector and where the absence of an edge…

If $G$ is a graph and $\vec H$ is an oriented graph, we write $G\to \vec H$ to say that every orientation of the edges of $G$ contains $\vec H$ as a subdigraph. We consider the case in which $G=G(n,p)$, the binomial random graph. We…

Motivated by the study of hammock (aka brick-wall) networks, we introduce in this paper the notion of X-path. Using the Jordan Curve Theorem for piecewise smooth curves, we prove duality properties for hammock networks. Consequences for…

Networks are inherently vulnerable to vertex failures, making the analysis of their structural robustness a fundamental problem in graph theory. In this study, we investigate the closeness and vertex residual closeness of graphs, with a…

Previously, the graph permanent was introduced as a single-valued invariant for graphs $G$ with $|E(G)| = k(|V(G)|-1)$ for some $k \in \mathbb{Z}_{>0}$. Herein, we construct the extended graph permanent, an infinite sequence for all graphs.…