Related papers: Maximal and Maximum Dissociation Sets in General a…

An $({\cal I},{\cal F}_d)$-partition of a graph is a partition of the vertices of the graph into two sets $I$ and $F$, such that $I$ is an independent set and $F$ induces a forest of maximum degree at most $d$. We show that for all $M<3$…

A triangle decomposition of a graph is a partition of its edges into triangles. A fractional triangle decomposition of a graph is an assignment of a non-negative weight to each of its triangles such that the sum of the weights of the…

For a fixed graph G, a maximal independent set is an independent set that is not a proper subset of any other independent set. P. Erd\"os, and independently, J. W. Moon and L. Moser, and R. E. Miller and D. E. Muller, determined the maximum…

For a graph $G$, a vertex subset $S$ is called a maximum generalized $k$-independent set if the induced subgraph $G[S]$ does not contain a $k$-tree as its subgraph, and the subset has maximum cardinality. The generalized $k$-independence…

A \emph{hole} in a graph is an induced cycle with at least 4 vertices. A graph is \emph{even-hole-free} if it does not contain a hole on an even number of vertices. A \emph{pyramid} is a graph made of three chordless paths $P_1 = a \dots…

The separation dimension of a graph $G$ is the smallest natural number $k$ for which the vertices of $G$ can be embedded in $\mathbb{R}^k$ such that any pair of disjoint edges in $G$ can be separated by a hyperplane normal to one of the…

Recently, settling a question of Erd\H{o}s, Balogh and Pet\v{r}\'{i}\v{c}kov\'{a} showed that there are at most $2^{n^2/8+o(n^2)}$ $n$-vertex maximal triangle-free graphs, matching the previously known lower bound. Here we characterize the…

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$. A set $I_0(G) \subseteq V(G)$ is a vertex independent set if no two vertices in $I_0(G)$ are adjacent in $G$. We study $\alpha_1(G)$, which is the maximum cardinality of a set…

An independent vertex set of a graph is a set of vertices of the graph in which no two vertices are adjacent, and a maximal independent set is one that is not a proper subset of any other independent set. In this paper we count the number…

The \emph{domination subdivision number} sd$(G)$ of a graph $G$ is the minimum number of edges that must be subdivided (where an edge can be subdivided at most once) in order to increase the domination number of $G$. It has been shown…

Let $\gamma(G)$ denote the domination number of a graph $G$. A vertex $v\in V(G)$ is called a \emph{critical vertex} of $G$ if $\gamma(G-v)=\gamma(G)-1$. A graph is called \emph{vertex-critical} if every vertex of it is critical. In this…

An independent set $I$ in a graph $G$ is maximal if $I$ is not properly contained in any other independent set of $G$. The study of maximal independent sets (MIS's) in various graphs is well-established, often focusing upon enumeration of…

An isolating set in a graph $G$ is a set $S$ of vertices such that removing $S$ and its neighborhood leaves no edge. The isolation number $\iota(G)$ of $G$ (also known as the vertex-edge domination number) is the minimum size among all…

An induced matching in a graph is a set of edges whose endpoints induce a $1$-regular subgraph. Gupta et al. (2012,\cite{Gupta}) showed that every $n$-vertex graph has at most $10^{\frac{n}{5}}\approx 1.5849^n$ maximal induced matchings,…

A \emph{complete geometric graph} consists of a set $P$ of $n$ points in the plane, in general position, and all segments (edges) connecting them. It is a well known question of Bose, Hurtado, Rivera-Campo, and Wood, whether there exists a…

We prove constructively that the maximum possible number of minimal connected dominating sets in a connected undirected graph of order $n$ is in $\Omega(1.489^n)$. This improves the previously known lower bound of $\Omega(1.4422^n)$ and…

Many extremal problems for graphs have threshold graphs as their extremal examples. For instance the current authors proved that for fixed $k\ge 1$, among all graphs on $n$ vertices with $m$ edges, some threshold graph has the fewest…

A set of vertices $S$ \emph{resolves} a graph $G$ if every vertex is uniquely determined by its vector of distances to the vertices in $S$. The \emph{metric dimension} of a graph $G$ is the minimum cardinality of a resolving set. In this…

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…

The \emph{regular independence number}, introduced by Albertson and Boutin in 1990, is the maximum cardinality of an independent set of $G$ in which all vertices have equal degree in $G$. Recently, Caro, Hansberg and Pepper introduced the…