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A vertex subset $S$ of a graph $G$ is a general position set of $G$ if no vertex of $S$ lies on a geodesic between two other vertices of $S$. The cardinality of a largest general position set of $G$ is the general position number…

In this work, we study two fundamental graph optimization problems, minimum vertex cover (MVC) and maximum-cardinality matching (MCM), for intersection graphs of geometric objects, e.g., disks, rectangles, hypercubes, etc., in…

A set S of vertices of a graph G is a geodesic transversal of G if every maximal geodesic of G contains at least one vertex of S. We determine a smallest geodesic transversal in certain interconnection networks such as mesh of trees, and…

A symmetric $m\times m$ matrix $M$ with entries taken from $\{0,1,\ast\}$ gives rise to a graph partition problem, asking whether a graph can be partitioned into $m$ vertex sets matched to the rows (and corresponding columns) of $M$ such…

A vertex $v\in V$ is said to resolve two vertices $x$ and $y$ if $d_G(v,x)\ne d_G(v,y)$. A set $S\subset V$ is said to be a metric generator for $G$ if any pair of vertices of $G$ is resolved by some element of $S$. A minimum metric…

A set D of vertices of a graph G with vertex set V is irredundant if each non-isolated vertex of G[D] has a neighbour in V-D that is not adjacent to any other vertex in D. The upper irredundance number IR(G) is the largest cardinality of an…

The general position problem for graphs was inspired by the no-three-in-line problem from discrete geometry. A set $S$ of vertices of a graph $G$ is a \emph{general position set} if no shortest path in $G$ contains three or more vertices of…

A set $X$ in the Euclidean space $\mathbb{R}^d$ is called an $m$-distance set if the set of Euclidean distances between two distinct points in $X$ has size $m$. An $m$-distance set $X$ in $\mathbb{R}^d$ is said to be maximal if there does…

M{\o}ller maps are identifications between the observables of a perturbatively interacting physical system and the observables of its underlying free (i.e. non-interacting) system. This work studies and characterizes obstructions to the…

Let $G = (A \cup B,E)$ be a bipartite graph where the set $A$ consists of agents or main players and the set $B$ consists of jobs or secondary players. Every vertex has a strict ranking of its neighbors. A matching $M$ is popular if for any…

A metric space $\mathrm{M}=(M,\de)$ is {\em indivisible} if for every colouring $\chi: M\to 2$ there exists $i\in 2$ and a copy $\mathrm{N}=(N, \de)$ of $\mathrm{M}$ in $\mathrm{M}$ so that $\chi(x)=i$ for all $x\in N$. The metric space…

A graph $G$ with four or more vertices is called bicritical if the removal of any pair of distinct vertices of $G$ results in a graph with a perfect matching. A bicritical graph is minimal if the deletion of each edge results in a…

A bipartite graph $G=(A,B,E)$ is ${\cal H}$-convex, for some family of graphs ${\cal H}$, if there exists a graph $H\in {\cal H}$ with $V(H)=A$ such that the set of neighbours in $A$ of each $b\in B$ induces a connected subgraph of $H$.…

Let $G$ be a finite group. A faithful $G$-variety $X$ is called strongly incompressible if every dominant $G$-equivariant rational map of $X$ onto another faithful $G$-variety $Y$ is birational. We settle the problem of existence of…

Let $G$ be a graph. A set $S \subseteq V(G)$ is independent if its elements are pairwise non-adjacent. A vertex $v \in V(G)$ is shedding if for every independent set $S \subseteq V(G) \setminus N[v]$ there exists $u \in N(v)$ such that $S…

In a graph $G$, let $\mu_G(xy)$ denote the number of edges between $x$ and $y$ in $G$. Let $\lambda K_{v,u}$ be the graph $(V\cup U,E)$ with $|V|=v$, $|U|=u$, and \[ \mu_G(xy)=\begin{cases} \lambda &\mbox{if $x\in U$ and $y\in V$ or if…

Two vertices $u, v \in V$ of an undirected connected graph $G=(V,E)$ are resolved by a vertex $w$ if the distance between $u$ and $w$ and the distance between $v$ and $w$ are different. A set $R \subseteq V$ of vertices is a $k$-resolving…

Let $G$ be a simple graph of order $n$. The double vertex graph $F_2(G)$ of $G$ is the graph whose vertices are the $2$-subsets of $V(G)$, where two vertices are adjacent in $F_2(G)$ if their symmetric difference is a pair of adjacent…

A subset $S$ of vertices of a connected graph $G$ is a distance-equalizer set if for every two distinct vertices $x, y \in V (G) \setminus S$ there is a vertex $w \in S$ such that the distances from $x$ and $y$ to $w$ are the same. The…

We show a general approach for detecting genuine multipartite entanglement (GME) and partial inseparability in many-body-systems by means of macroscopic observables (such as the energy) only. We show that the obtained criteria, the "GME…