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We introduce a novel lower bound technique for distributed graph algorithms under bandwidth limitations. We define the notion of \emph{fooling views} and exemplify its strength by proving two new lower bounds for triangle membership in the…
Erd\H{o}s asked whether for any $n$-vertex graph $G$, the parameter $p^*(G)=\min \sum_{i\ge 1} (|V(G_i)|-1)$ is at most $\lfloor n^2/4\rfloor$, where the minimum is taken over all edge decompositions of $G$ into edge-disjoint cliques $G_i$.…
The \emph{metric dimension} of a graph $G$, denoted by $\dim(G)$, is the minimum number of vertices such that each vertex is uniquely determined by its distances to the chosen vertices. Let $G_1$ and $G_2$ be disjoint copies of a graph $G$…
A mobile agent, modeled as a deterministic finite automaton, navigates in the infinite anonymous oriented grid $\mathbb{Z} \times \mathbb{Z}$. It has to explore a given infinite subgraph of the grid by visiting all of its nodes. We focus on…
The game of peg solitaire on graphs was introduced by Beeler and Hoilman in 2011. In this game, pegs are initially placed on all but one vertex of a graph $G$. If $xyz$ forms a path in $G$ and there are pegs on vertices $x$ and $y$ but not…
Computing the number of realizations of a minimally rigid graph is a notoriously difficult problem. Towards this goal, for graphs that are minimally rigid in the plane, we take advantage of a recently published algorithm, which is the…
We define a new graph invariant called the scramble number. We show that the scramble number of a graph is a lower bound for the gonality and an upper bound for the treewidth. Unlike the treewidth, the scramble number is not minor monotone,…
On a connected finite graph, we propose an evolution of weights including Ollivier's Ricci flow as a special case. During the evolution process, on each edge, the speed of change of weight is exactly the difference between the Wasserstein…
We study a family of positive weighted well-covered graphs, which we call levelable graphs, that are related to a construction of level artinian rings in commutative algebra. A graph $G$ is levelable if there exists a weight function with…
We consider the graphs whose edges are marked by the integers (weights) from $0$ to $q-1$ (zero corresponds to no-edge). Such graph is called additive if its vertices can be marked in such a way that the weight of every edge is equal to the…
Given an input graph and weights on its vertices, the maximum co-2-plex problem is to find a subset of vertices maximizing the sum of their weights and inducing a graph of degree at most 1. In this article, we analyze polyhedral aspects of…
We construct a family of planar graphs $\{G_n\}_{n\geq 4}$, where $G_n$ has $n$ vertices including a source vertex $s$ and a sink vertex $t$, and edge weights that change linearly with a parameter $\lambda$ such that, as $\lambda$ varies in…
Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$, and let $d(u,w)$ denote the length of a $u-w$ geodesic in $G$. For any $v\in V(G)$ and $e=xy\in E(G)$, let $d(e,v)=\min\{d(x,v),d(y,v)\}$. For distinct $e_1, e_2\in E(G)$, let…
We report on the erosion of flat linoleum "pebbles" under steady rotation in a slurry of abrasive grit. To quantify shape as a function of time, we develop a general method in which the pebble is photographed from multiple angles with…
Exploratory analysis over network data is often limited by the ability to efficiently calculate graph statistics, which can provide a model-free understanding of the macroscopic properties of a network. We introduce a framework for…
Turbulence plays a key role in the transport of pebble-sized particles. It also affects the ability of pebbles to be accreted by protoplanets, because it stirs pebbles out of the disk midplane. In addition, turbulence can suppress pebble…
A straight-line drawing $\delta$ of a planar graph $G$ need not be plane, but can be made so by \emph{untangling} it, that is, by moving some of the vertices of $G$. Let shift$(G,\delta)$ denote the minimum number of vertices that need to…
The systole of a hyperbolic surface is bounded by a logarithmic function of its genus. This bound is sharp, in that there exist sequences of surfaces with genera tending to infinity that attain logarithmically large systoles. These are…
We consider two operations on an edge of an embedded graph (or equivalently a ribbon graph): giving a half-twist to the edge and taking the partial dual with respect to the edge. These two operations give rise to an action of S_3^{|E(G)|},…
A simple graph more often than not contains adjacent vertices with equal degrees. This in particular holds for all pairs of neighbours in regular graphs, while a lot such pairs can be expected e.g. in many random models. Is there a…