Related papers: A Sard theorem for graph theory

The \emph{difference subgroup graph} $D(G)$ of a finite group $G$ is defined as the graph whose vertices are the non-trivial proper subgroups of $G$, with two distinct vertices $H$ and $K$ adjacent if and only if $\langle H, K \rangle = G$…

The pinched sphere is the pseudo-surface $\mathbb{S}^{\circ}_0$ obtained by identifying two distinct points of the sphere. We provide a structural characterization of graphs excluding an $\mathbb{S}^{\circ}_0$-embeddable graph as a minor.…

In a series of three papers we develop an end space theory for directed graphs. As for undirected graphs, the ends of a digraph are points at infinity to which its rays converge. Unlike for undirected graphs, some ends are joined by limit…

The \emph{zero forcing number}, $Z(G)$, of a graph $G$ is the minimum cardinality of a set $S$ of black vertices (whereas vertices in $V(G) \setminus S$ are colored white) such that $V(G)$ is turned black after finitely many applications of…

A graph is $d$-rigid if for any generic realisation of the graph in $\mathbb{R}^d$ (equivalently, the $d$-dimensional sphere $\mathbb{S}^d$), there are only finitely many non-congruent realisations in the same space with the same edge…

Given two graphs $H_1$ and $H_2$, a graph $G$ is $(H_1,H_2)$-free if it contains no induced subgraph isomorphic to $H_1$ or $H_2$. Let $P_t$ be the path on $t$ vertices. A graph $G$ is $k$-vertex-critical if $G$ has chromatic number $k$ but…

The Gram dimension $\gd(G)$ of a graph is the smallest integer $k \ge 1$ such that, for every assignment of unit vectors to the nodes of the graph, there exists another assignment of unit vectors lying in $\oR^k$, having the same inner…

We prove a Lefschetz formula for general simple graphs which equates the Lefschetz number L(T) of an endomorphism T with the sum of the degrees i(x) of simplices in G which are fixed by T. The degree i(x) of x with respect to T is defined…

Graph modification problems are computational tasks where the goal is to change an input graph $G$ using operations from a fixed set, in order to make the resulting graph satisfy a target property, which usually entails membership to a…

Let $G$ be a simple graph with vertex set $V(G)$. A set $S\subseteq V(G)$ is independent if no two vertices from $S$ are adjacent. For $X\subseteq V(G)$, the difference of $X$ is $d(X) = |X|-|N(X)|$ and an independent set $A$ is critical if…

Let $G = (V, E)$ be a graph and $\lambda $ a non-negative integer. A graph $G$ is called a $(\lambda, 1)$-{\em graph} if $ (c0)$ $G$ is neither a complete graph no an edge-empty graph, $ (c1)$ every edge in $G$ belongs to exactly $\lambda$…

Let $G$ be a connected graph on $n$ vertices with adjacency matrix $A_G$. Associated to $G$ is a polynomial $d_G(x_1,\dots, x_n)$ of degree $n$ in $n$ variables, obtained as the determinant of the matrix $M_G(x_1,\dots,x_n)$, where…

For a graph $G$, the central graph $C(G)$ is the graph constructed from $G$ by subdividing each edge of $G$ with one vertex and also by adding an edge to every pair of non-adjacent vertices in $G$. Also for a graph $G$, let $\gamma(G)$ and…

Let $G$ be a simple graph with $n$ vertices and $m$ edges. According to Tur\'{a}n's theorem, if $G$ is $K_{r+1}$-free, then $m \leq |E(T(n, r))|,$ where $T(n, r)$ denotes the Tur\'{a}n graph on $n$ vertices with a maximum clique of order…

The emerging theory of graph limits exhibits an analytic perspective on graphs, showing that many important concepts and tools in graph theory and its applications can be described more naturally (and sometimes proved more easily) in…

We give a structural description of the class $\cal C$ of graphs that do not contain a cycle with a unique chord as an induced subgraph. Our main theorem states that any connected graph in $\cal C$ is either in some simple basic class or…

A weighted (directed) graph is a (directed) graph with integer weights assigned to its vertices and edges. The weight of a subgraph is the sum of weights of vertices and edges in the subgraph. The problem of determining the largest order…

Graphs triangulating the $2$-sphere are generically rigid in $3$-space, due to Gluck-Dehn-Alexandrov-Cauchy. We show there is a \emph{finite} subset $A$ in $3$-space so that the vertices of each graph $G$ as above can be mapped into $A$ to…

The twin-width of a graph $G$ is the minimum integer $d$ such that $G$ has a $d$-contraction sequence, that is, a sequence of $|V(G)|-1$ iterated vertex identifications for which the overall maximum number of red edges incident to a single…

The $k$-token graph $F_k(G)$ of a graph $G$ is the graph whose vertices are the $k$-subsets of vertices from $G$, two of which being adjacent whenever their symmetric difference is a pair of adjacent vertices in $G$. It is a known result…