Related papers: Edgeless graphs are the only universal fixers

A divisor graph $G$ is an ordered pair $(V, E)$ where $V \subset \mathbbm{Z}$ and for all $u \neq v \in V$, $u v \in E$ if and only if $u \mid v$ or $v \mid u$. A graph which is isomorphic to a divisor graph is also called a divisor graph.…

A graph is reconstructible if it is determined up to isomorphism from the collection of all its one-vertex-deleted subgraphs, known as the deck of G. The Reconstruction Conjecture (RC) posits that every finite simple graph with at least…

Given a graph $G$, two edges $e_{1},e_{2}\in E(G)$ are said to have a common edge $e$ if $e$ joins an endvertex of $e_{1}$ to an endvertex of $e_{2}$. A subset $B\subseteq E(G)$ is an edge open packing set in $G$ if no two edges of $B$ have…

A total weighting of a graph $G$ is a mapping $f$ which assigns to each element $z \in V(G) \cup E(G)$ a real number $f(z)$ as its weight. The vertex sum of $v$ with respect to $f$ is $\phi_f(v)=\sum_{e \in E(v)}f(e)+f(v)$. A total…

For a graph G, the k-total dominating graph D_{k}^{t}(G) is the graph whose vertices correspond to the total dominating sets of G that have cardinality at most k; two vertices of D_{k}^{t}(G) are adjacent if and only if the corresponding…

A subset $S$ of vertices in a graph $G$ is a secure total dominating set of $G$ if $S$ is a total dominating set of $G$ and, for each vertex $u \not\in S$, there is a vertex $v \in S$ such that $uv$ is an edge and $(S \setminus \{v\}) \cup…

A graph $G$ is $(k,k')$-choosable if the following holds: For any list assignment $L$ which assigns to each vertex $v$ a set $L(v)$ of $k$ real numbers, and assigns to each edge $e$ a set $L(e)$ of $k'$ real numbers, there is a total…

Let $G$ be a finite simple graph on the vertex set $V(G) = \{x_1, \ldots, x_n\}$ and $I(G) \subset K[V(G)]$ its edge ideal, where $K[V(G)]$ is the polynomial ring in $x_1, \ldots, x_n$ over a field $K$ with each ${\rm deg} x_i = 1$ and…

We investigate how to find generic and globally rigid realizations of graphs in $\mathbb{R}^d$ based on elementary geometric observations. Our arguments lead to new proofs of a combinatorial characterization of the global rigidity of graphs…

A graph automorphism is a bijective mapping of the vertices that preserves adjacent vertices. A vertex determining set of a graph is a set of vertices such that the only automorphism that fixes those vertices is the identity. The size of a…

In Partition Into Complementary Subgraphs (Comp-Sub) we are given a graph $G=(V,E)$, and an edge set property $\Pi$, and asked whether $G$ can be decomposed into two graphs, $H$ and its complement $\overline{H}$, for some graph $H$, in such…

Any graph which is not vertex transitive has a proper induced subgraph which is unique due to its structure or the way of its connection to the rest of the graph. We have called such subgraph as an anchor. Using an anchor which, in fact, is…

A graph $G$ is said to be super edge-magic if there exists a bijective function $f:V\left(G\right) \cup E\left(G\right)\rightarrow \left\{1, 2, \ldots , \left\vert V\left( G\right) \right\vert +\left\vert E\left( G\right) \right\vert…

A path in an edge-colored graph is called proper if no two consecutive edges of the path receive the same color. For a connected graph $G$, the proper connection number $pc(G)$ of $G$ is defined as the minimum number of colors needed to…

A dominating set in a graph $G$ is a set $S$ of vertices such that every vertex in $V(G) \setminus S$ is adjacent to a vertex in $S$. A restrained dominating set of $G$ is a dominating set $S$ with the additional restraint that the graph $G…

In this paper, we study commutative zero-divisor semigroups determined by graphs. We prove a uniqueness theorem for a class of graphs. We show two classes of graphs that have no corresponding semigroups. In particular, any complete graph…

An immersion of a graph H in another graph G is a one-to-one mapping phi:V(H)->V(G) and a collection of edge-disjoint paths in G, one for each edge of H, such that the path P_{uv} corresponding to the edge uv has endpoints phi(u) and…

The distinguishing index $D'(G)$ of a graph $G$ is the least number of colours needed in an edge colouring which is not preserved by any non-trivial automorphism. Broere and Pil\'sniak conjectured that if every non-trivial automorphism of a…

We investigate which planar point sets allow simultaneous straight-line embeddings of all planar graphs on a fixed number of vertices. We first show that $(1.293-o(1))n$ points are required to find a straight-line drawing of each $n$-vertex…

A simple topological graph T = (V(T), E(T)) is a drawing of a graph in the plane where every two edges have at most one common point (an endpoint or a crossing) and no three edges pass through a single crossing. Topological graphs G and H…