Related papers: Packing independent cliques into planar graphs
In this paper, we give the sharp upper bound for the number of vertices with positive curvature in a planar graph with nonnegative combinatorial curvature. Based on this, we show that the automorphism group of a planar---possibly…
In this paper, we relate the problem of finding a maximum clique to the intersection number of the input graph (i.e. the minimum number of cliques needed to edge cover the graph). In particular, we consider the maximum clique problem for…
Finding the maximum clique is a known NP-Complete problem and it is also hard to approximate. This work proposes two efficient algorithms to obtain it. Nevertheless, the first one is able to fins the maximum for some special cases, while…
We show that the number of independent sets in an N-vertex, d-regular graph is at most (2^{d+1} - 1)^{N/2d}, where the bound is sharp for a disjoint union of complete d-regular bipartite graphs. This settles a conjecture of Alon in 1991 and…
The $k$-independence number of a graph, $\alpha_k(G)$, is the maximum size of a set of vertices at pairwise distance greater than $k$, or alternatively, the independence number of the $k$-th power graph $G^k$. Although it is known that…
We study the problem of maximizing the number of independent sets in $n$-vertex $k$-chromatic $\ell$-connected graphs. First we consider maximizing the total number of independent sets in such graphs with $n$ sufficiently large, and for…
The subgraph number of a vertex in a graph is defined as the number of connected subgraphs containing that vertex. The graph and its vertex which correspond to the minimum subgraph number among all graphs on $n$ vertices and $k$ cut…
Define a boundary point of a graph which is embedded in the Euclidean plane a vertex which is incident to only one edge. In this paper we consider graphs which are embedded in the Euclidean plane with a finite number of boundary points. The…
A vertebrate interval graph is an interval graph in which the maximum size of a set of independent vertices equals the number of maximal cliques. For any fixed $v \ge 1$, there is a polynomial-time algorithm for deciding whether a…
A set $B$ of vertices in a graph $G$ is called a \emph{$k$-limited packing} if for each vertex $v$ of $G$, its closed neighbourhood has at most $k$ vertices in $B$. The \emph{$k$-limited packing number} of a graph $G$, denoted by $L_k(G)$,…
We describe an infinite family of graphs $G_n$, where $G_n$ has $n$ vertices, independence number at least $n/4$, and no set of less than $\sqrt{n}/2$ vertices intersects all its maximum independent sets. This is motivated by a question of…
The decycling number of a graph $G$ is the minimum number of vertices whose removal from $G$ results in an acyclic subgraph. It is known that determining the decycling number of a graph $G$ is equivalent to finding the maximum induced…
A set $S$ of vertices in a graph is an open packing if (open) neighborhoods of any two distinct vertices in $S$ are disjoint. In this paper, we consider the graphs that have a unique maximum open packing. We characterize the trees with this…
We show that the edges of any planar graph of maximum degree at most $9$ can be partitioned into $4$ linear forests and a matching. Combined with known results, this implies that the edges of any planar graph $G$ of odd maximum degree…
We derive upper and lower bounds for the vertex-isoperimetric number of the incidence graphs of unitals and determine its order of magnitude. In the case when a unital contains sufficiently large arcs, these bounds agree and give rise to…
In the well known planted clique problem, a clique (or alternatively, an independent set) of size $k$ is planted at random in an Erdos-Renyi random $G(n, p)$ graph, and the goal is to design an algorithm that finds the maximum clique (or…
We study the existence and the number of $k$-dominating independent sets in certain graph families. While the case $k=1$ namely the case of maximal independent sets - which is originated from Erd\H{o}s and Moser - is widely investigated,…
Given a directed graph, an acyclic set is a set of vertices inducing a subgraph with no directed cycle. In this note we show that there exist oriented planar graphs of order $n$ for which the size of the maximum acyclic set is at most…
Given a finite nonempty sequence $S$ of integers, write it as $XY^k$, consisting of a prefix $X$ (which may possibly be empty), followed by $k$ copies of a non-empty string $Y$. Then, the greatest such integer $k$ is called the curling…
A subset of vertices is called a dissociation set if it induces a subgraph with vertex degree at most one. Recently, Yuan et al. established the upper bound of the maximum number of dissociation sets among all connected graphs of order n…