Related papers: Nullstellensatz Size-Degree Trade-offs from Revers…
A pebbling move on a graph consists of taking two pebbles off from one vertex and add one pebble on an adjacent vertex, the $t$-pebbling number of a graph $G$ is the minimum number of pebbles so that we can move $t$ pebbles on any vertex on…
Graph pebbling is a game played on graphs with pebbles on their vertices. A pebbling move removes two pebbles from one vertex and places one pebble on an adjacent vertex. The pebbling number is the smallest $t$ so that from any initial…
The \textit{node reliability} of a graph $G$ is the probability that at least one node is operational and that the operational nodes can all communicate in the subgraph that they induce, given that the edges are perfectly reliable but each…
Let $\mathcal{Q}$ be a vertex subset problem on graphs. In a reconfiguration variant of $\mathcal{Q}$ we are given a graph $G$ and two feasible solutions $S_s, S_t\subseteq V(G)$ of $\mathcal{Q}$ with $|S_s|=|S_t|=k$. The problem is to…
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…
We give a general method for producing various effective Null and Positivstellens\"atze, and getting new Positivstellens\"atze in algebraically closed valued fields and ordered groups. These various effective Nullstellens\"atze produce…
This paper begins the study of reconfiguration of zero forcing sets, and more specifically, the zero forcing graph. Given a base graph $G$, its zero forcing graph, $\mathscr{Z}(G)$, is the graph whose vertices are the minimum zero forcing…
Determining the size of a maximum independent set of a graph $G$, denoted by $\alpha(G)$, is an NP-hard problem. Therefore, many attempts are made to find upper and lower bounds, or exact values of $\alpha (G)$ for special classes of…
The random reversal graph offers new perspectives, allowing to study the connectivity of genomes as well as their most likely distance as a function of the reversal rate. Our main result shows that the structure of the random reversal graph…
We show how to assign labels of size $\tilde O(1)$ to the vertices of a directed planar graph $G$, such that from the labels of any three vertices $s,t,f$ we can deduce in $\tilde O(1)$ time whether $t$ is reachable from $s$ in the graph…
We prove a noncommutative real Nullstellensatz for 2-step nilpotent Lie algebras that extends the classical, commutative real Nullstellensatz as follows: Instead of the real polynomial algebra $\mathbb R[x_1, \dots, x_d]$ we consider the…
In this paper we construct a $(2,2)$ dimensional string theory with manifest $N=1$ spacetime supersymmetry. We use Berkovits' approach of augmenting the spacetime supercoordinates by the conjugate momenta for the fermionic variables. The…
Let $G$ be a $3$-connected ordered graph with $n$ vertices and $m$ edges. Let $\mathbf{p}$ be a randomly chosen mapping of these $n$ vertices to the integer range $\{1, 2,3, \ldots, 2^b\}$ for $b\ge m^2$. Let $\ell$ be the vector of $m$…
Let $G$ be a connected graph. A pebbling move is defined as taking two pebbles from one vertex and placing one pebble to an adjacent vertex and throwing away the other pebble. The non-split domination cover pebbling number, $\psi_{ns}(G)$,…
In graph realization problems one is given a degree sequence and the task is to decide whether there is a graph whose vertex degrees match to the given sequence. This realization problem is known to be polynomial-time solvable when the…
In this note we generalize a result from a recent paper of Hajac, Reznikoff and Tobolski (2020). In that paper they give conditions they call admissibility on a pushout diagram in the category of directed graphs implying that the…
We discuss here some computational aspects of the Combinatorial Nullstellensatz argument. Our main result shows that the order of magnitude of the symmetry group associated with permutations of the variables in algebraic constraints,…
We study the Borel-reducibility of isomorphism relations of complete first order theories and show the consistency of the following: For all such theories T and T', if T is classifiable and T' is not, then the isomorphism of models of T' is…
In a graph $G$, we define a set of vertices to be a \emph{strong hub set} if for any two vertices in $G$, we can find a path between them whose internal vertices are all in this set. We define the \emph{strong hub cover pebbling number} of…
A pebbling move on a graph removes two pebbles from a vertex and adds one pebble to an adjacent vertex. A vertex is reachable from a pebble distribution if it is possible to move a pebble to that vertex using pebbling moves. The optimal…