Related papers: Graphs with a unique maximum open packing
A graph is well-covered if all its maximal independent sets are of the same size (M. D. Plummer, 1970). A well-covered graph is 1-well-covered if the deletion of every vertex leaves a graph which is well-covered as well (J. W. Staples,…
Let $G=(V,E)$ be an undirected graph without loops and multiple edges. A subset $C\subseteq V$ is called \emph{identifying} if for every vertex $x\in V$ the intersection of $C$ and the closed neighbourhood of $x$ is nonempty, and these…
Let $k\ge 2$ be an integer and $T_1,\ldots, T_k$ be spanning trees of a graph $G$. If for any pair of vertices $(u,v)$ of $V(G)$, the paths from $u$ to $v$ in each $T_i$, $1\le i\le k$, do not contain common edges and common vertices,…
Given $D$ and $\gamma>0$, whenever $c>0$ is sufficiently small and $n$ sufficiently large, if $\mathcal{G}$ is a family of $D$-degenerate graphs of individual orders at most $n$, maximum degrees at most $\tfrac{cn}{\log n}$, and total…
For a fixed graph H with t vertices, an H-factor of a graph G with n vertices, where t divides n, is a collection of vertex disjoint (not necessarily induced) copies of H in G covering all vertices of G. We prove that for a fixed tree T on…
We prove that one can perfectly pack degenerate graphs into complete or dense $n$-vertex quasirandom graphs, provided that all the degenerate graphs have maximum degree $o(\frac{n}{\log n})$, and in addition $\Omega(n)$ of them have at most…
In the Independent set problem, the input is a graph $G$, every vertex has a non-negative integer weight, and the task is to find a set $S$ of pairwise non-adjacent vertices, maximizing the total weight of the vertices in $S$. We give an…
In the Matching Cut problem we ask whether a graph $G$ has a matching cut, that is, a matching which is also an edge cut of $G$. We consider the variants Perfect Matching Cut and Disconnected Perfect Matching where we ask whether there…
Let $S$ be an independent set of a simple undirected graph $G$. Suppose that each vertex of $S$ has a token placed on it. The tokens are allowed to be moved, one at a time, by sliding along the edges of $G$, so that after each move, the…
An instance of the graph-constrained max-cut (GCMC) problem consists of (i) an undirected graph G and (ii) edge-weights on a complete undirected graph on the same vertex set. The objective is to find a subset of vertices satisfying some…
The recently introduced $\{k\}$-packing function problem is considered in this paper. Special relation between a case when $k=1$, $k\ge 2$ and linear programming relaxation is introduced with sufficient conditions for optimality. For…
Tree-decompositions of graphs are of fundamental importance in structural and algorithmic graph theory. The main property of tree-decompositions is the width (the maximum size of a bag minus 1). We show that every graph has a…
The K-way vertex cut problem} consists in, given a graph G, finding a subset of vertices of a given size, whose removal partitions G into the maximum number of connected components. This problem has many applications in several areas. It…
Pairwise compatibility graphs (PCGs) with non-negative integer edge weights recently have been used to describe rare evolutionary events and scenarios with horizontal gene transfer. Here we consider the case that vertices are separated by…
Two permutations of the vertices of a graph $G$ are called $G$-different if there exists an index $i$ such that $i$-th entry of the two permutations form an edge in $G$. We bound or determine the maximum size of a family of pairwise…
In the Maximum Independent Set problem we are asked to find a set of pairwise nonadjacent vertices in a given graph with the maximum possible cardinality. In general graphs, this classical problem is known to be NP-hard and hard to…
The Near-Bipartiteness problem is that of deciding whether or not the vertices of a graph can be partitioned into sets $A$ and $B$, where $A$ is an independent set and $B$ induces a forest. The set $A$ in such a partition is said to be an…
We find the maximum number of maximal independent sets in two families of graphs: all graphs with $n$ vertices and at most $r$ cycles, and all such graphs that are also connected. In addition, we characterize the extremal graphs.
In a graph, a matching cut is an edge cut that is a matching. Matching Cut is the problem of deciding whether or not a given graph has a matching cut, which is known to be NP-complete even when restricted to bipartite graphs. It has been…
Two graphs $G_1$ and $G_2$ on $n$ vertices are said to pack if there exist injective mappings of their vertex sets into $[n]$ such that the images of their edge sets are disjoint. A longstanding conjecture due to Bollob\'as and Eldridge…