Related papers: Edgeless graphs are the only universal fixers

Let alpha(G) denote the maximum size of an independent set of vertices and mu(G) be the cardinality of a maximum matching in a graph G. A matching saturating all the vertices is perfect. If alpha(G) + mu(G) equals the number of vertices of…

A planar graph $G$ is said to be non-separating if there exists an embedding of $G$ in $\mathbb{R}^2$ such that for any cycle $\mathcal{C}\subset G$, all vertices of $G\setminus \mathcal{C}$ are within the same connected component of…

It has been conjectured that for every claw-free graph $G$ the choice number of $G$ is equal to its chromatic number. We focus on the special case of this conjecture where $G$ is perfect. Claw-free perfect graphs can be decomposed via…

For a simple graph $G=(V,E)$, a \emph{proper total weighting} is a mapping $w: V\cup E\rightarrow \mathbb R$ such that for every edge $uv\in E$, $w(u)+\sum_{e\ni u}w(e)\neq w(v)+\sum_{e\ni v}w(e)$. The graph $G$ is said…

A subset $D\subseteq V_G$ is a dominating set of $G$ if every vertex in $V_G\setminus D$ has a neighbor in $D$, while $D$ is a 2-dominating set of $G$ if every vertex belonging to $V_G\setminus D$ is joined by at least two edges with a…

A graph is perfectly divisible if for each of its induced subgraph $H$, $V(H)$ can be partitioned into $A$ and $B$ such that $H[A]$ is perfect and $\omega(H[B]) < \omega(H)$. A graph $G$ is perfectly weight divisible if for every positive…

It is shown, for a given graph group $G$, that the fixed point subgroup Fix$\,\varphi$ is finitely generated for every endomorphism $\varphi$ of $G$ if and only if $G$ is a free product of free abelian groups. The same conditions hold for…

Let $G = (V,E)$ be a simple, undirected and connected graph. A connected (total) dominating set $S \subseteq V$ is a secure connected (total) dominating set of $G$, if for each $ u \in V \setminus S$, there exists $v \in S$ such that $uv…

We exactly settle the complexity of graph realization, graph rigidity, and graph global rigidity as applied to three types of graphs: "globally noncrossing" graphs, which avoid crossings in all of their configurations; matchstick graphs,…

Circular arc graphs are graphs whose vertices can be represented as arcs on a circle such that any two vertices are adjacent if and only if their corresponding arcs intersect. Proper circular arc graphs are graphs which have a circular arc…

A graph is one-ended if it contains a ray (a one way infinite path) and whenever we remove a finite number of vertices from the graph then what remains has only one component which contains rays. A vertex $v$ {\em dominates} a ray in the…

We explore a reconfiguration version of the dominating set problem, where a dominating set in a graph $G$ is a set $S$ of vertices such that each vertex is either in $S$ or has a neighbour in $S$. In a reconfiguration problem, the goal is…

Let $G_1$ and $G_2$ be disjoint copies of a graph $G$, and let $f: V(G_1) \rightarrow V(G_2)$ be a function. Then a \emph{functigraph} $C(G, f)=(V, E)$ has the vertex set $V=V(G_1) \cup V(G_2)$ and the edge set $E=E(G_1) \cup E(G_2) \cup…

A \emph{self-complementary} graph is a graph isomorphic to its complement. An isomorphism between $G$ and its complement, viewed as a permutation of $V(G)$, is then called an \emph{antimorphism}. A \emph{skew partition} of $G$ is a…

Let $G=(V,E)$ be a simple graph. A dominating set of $G$ is a subset $S\subseteq V$ such that every vertex not in $S$ is adjacent to at least one vertex in $S$. The cardinality of a smallest dominating set of $G$, denoted by $\gamma(G)$, is…

Berge Conjecture states that every bridgeless cubic graph has 5 perfect matchings such that each edge is contained in at least one of them. In this paper, we show that Berge Conjecture holds for two classes of cubic graphs, cubic graphs…

A set $S$ of vertices in a graph $G$ is a dominating set if every vertex not in $S$ is adjacent to a vertex in $S$. If, in addition, $S$ is an independent set, then $S$ is an independent dominating set. The independent domination number…

A dominating set D in a graph G is a subset of its vertices such that every vertex of the graph which does not belong to set D is adjacent to at least one vertex from set D. A set of vertices of graph G is a global dominating set if it is a…

Let $G$ be a finite non-solvable group with solvable radical $Sol(G)$. The solvable graph $\Gamma_s(G)$ of $G$ is a graph with vertex set $G\setminus Sol(G)$ and two distinct vertices $u$ and $v$ are adjacent if and only if $\langle u, v…

A dominating set of a graph $G$ is a set $D\subseteq V(G)$ such that \-every vertex of $G$ is either in $D$ or is adjacent to a vertex in $D$. The domination number of $G$, $\gamma(G)$, is the minimum order of a dominating set. A subset $R$…