Related papers: Characterizing globally linked pairs in graphs

We consider the rigidity and global rigidity of bar-joint frameworks in Euclidean $d$-space under additional dilation constraints in specified coordinate directions. In this setting we obtain a complete characterisation of generic rigidity.…

Let $G$ be a graph with $n$ vertices, and $d$ be a target dimension. In this paper we study the set of rank $n-d-1$ matrices that are equilibrium stress matrices for at least one (unspecified) $d$-dimensional framework of $G$ in general…

We consider the problem of characterising the generic rigidity of bar-joint frameworks in $\mathbb{R}^d$ in which each vertex is constrained to lie in a given affine subspace. The special case when $d=2$ was previously solved by I. Streinu…

Let $G=(V,E)$ be a graph. A subset $S \subseteq V$ is called a global dominating set of $G$, if it serves as a dominating set in both $G$ and its complement $\overline{G}$. We define two disjoint subsets $V_1,V_2 \subseteq V$ to form a…

Let $G$ be a connected graph with vertex set $V(G)=\{v_1,v_2,...,v_{\nu}\}$, which may have multiple edges but have no loops, and $2\leq d_G(v_i)\leq 3$ for $i=1,2,...,\nu$, where $d_G(v)$ denotes the degree of vertex $v$ of $G$. We show…

For a simple graph $G$, the $2$-distance graph, $D_2(G)$, is a graph with the vertex set $V(G)$ and two vertices are adjacent if and only if their distance is $2$ in the graph $G$. In this paper, we characterize all graphs with connected…

Let $k \geq 1$. A graph $G$ is $\mathbf{W_k}$ if for any $k$ pairwise disjoint independent vertex subsets $A_1, \dots, A_k$ in $G$, there exist $k$ pairwise disjoint maximum independent sets $S_1, \dots, S_k$ in $G$ such that $A_i \subseteq…

A $d$-dimensional tensegrity framework $(T,p)$ is an edge-labeled geometric graph in ${\mathbb R}^d$, which consists of a graph $T=(V,B\cup C\cup S)$ and a map $p:V\to {\mathbb R}^d$. The labels determine whether an edge $uv$ of $T$…

A directed graph $G=(V,E)$ is called strongly biconnected if $G$ is strongly connected and the underlying graph of $G$ is biconnected. A strongly biconnected component of a strongly connected graph $G=(V,E)$ is a maximal vertex subset…

In 1959, Erd\H{o}s and Gallai showed that every $2$-connected graph $G$ contains a cycle of length at least $\frac{2|E(G)|}{|V(G)|-1}$. This result was subsequently extended to weighted graphs by Bondy and Fan in 1991. A natural local…

For any natural number $d$, a graph $G$ is a (disjoint) $d$-interval graph if it is the intersection graph of (disjoint) $d$-intervals, the union of $d$ (disjoint) intervals on the real line. Two important subclasses of $d$-interval graphs…

A greedy embedding of a graph $G = (V,E)$ into a metric space $(X,d)$ is a function $x : V(G) \to X$ such that in the embedding for every pair of non-adjacent vertices $x(s), x(t)$ there exists another vertex $x(u)$ adjacent to $x(s)$ which…

Suppose a finite, unweighted, combinatorial graph $G = (V,E)$ is the union of several (degree-)regular graphs which are then additionally connected with a few additional edges. $G$ will then have only a small number of vertices $v \in V$…

A graph $H$ is said to be positive if the homomorphism density $t_H(G)$ is non-negative for all weighted graphs $G$. The positive graph conjecture proposes a characterisation of such graphs, saying that a graph is positive if and only if it…

A graph $G$ is $[a,b]$-covered if for each edge $e$ of $G$ there is an $[a,b]$-factor containing it. For $a=b=1$, an $[a,b]$-covered graph is a matching covered graph. The structural theory of matching covered graphs constitutes a…

A graph is said to be globally rigid in $d$-dimensional space if almost all of its embeddings are unique up to isometries. If a graph has enough automorphisms to send any of its vertices into any other, then it is called vertex-transitive.…

We prove that every (6k + 2l, 2k)-connected simple graph contains k rigid and l connected edge-disjoint spanning subgraphs. This implies a theorem of Jackson and Jord\'an [4] and a theorem of Jord\'an [6] on packing of rigid spanning…

Tibor Gallai conjectured that the edge set of every connected graph $G$ on $n$ vertices can be partitioned into $\lceil n/2\rceil$ paths. Let $\mathcal{G}_{k}$ be the class of all $2k$-regular graphs of girth at least $2k-2$ that admit a…

It is shown that for any outerplanar graph G there is a one to one mapping of the vertices of G to the plane, so that the number of distinct distances between pairs of connected vertices is at most three. This settles a problem of Carmi,…

A set $D$ of vertices of a graph $G$ is locating if every two distinct vertices outside $D$ have distinct neighbors in $D$; that is, for distinct vertices $u$ and $v$ outside $D$, $N(u) \cap D \neq N(v) \cap D$, where $N(u)$ denotes the…