Related papers: Equality of Distance Packing Numbers

Reconfiguration problems ask whether one feasible solution can be transformed into another by a sequence of local moves while maintaining feasibility throughout. For integers $d \geq 1$ and $k \geq d+1$, the Distance Coloring problem asks…

A packing $k$-coloring of a graph $G$ is a partition of $V(G)$ into sets $V_1,\ldots,V_k$ such that for each $1\leq i\leq k$ the distance between any two distinct $x,y\in V_i$ is at least $i+1$. The packing chromatic number, $\chi_p(G)$, of…

Motivated by applications in graph drawing and information visualization, we examine the planar split thickness of a graph, that is, the smallest $k$ such that the graph is $k$-splittable into a planar graph. A $k$-split operation…

For every probability $p\in[0,1]$ we define a distance-based graph property, the $p$TS-distance-balancedness, that in the case $p=0$ coincides with the standard distance-balancedness, and in the case $p=1$ is related to the…

Some graphs admit drawings in the Euclidean k-space in such a (natu- ral) way, that edges are represented as line segments of unit length. Such drawings will be called k dimensional unit distance representations. When two non-adjacent…

It is shown in this note that approximating the number of independent sets in a $k$-uniform linear hypergraph with maximum degree at most $\Delta$ is NP-hard if $\Delta\geq 5\cdot 2^{k-1}+1$. This confirms that for the relevant sampling and…

Simplifying graphs is a very applicable problem in numerous domains, especially in computational geometry. Given a geometric graph and a threshold, the minimum-complexity graph simplification asks for computing an alternative graph of…

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…

A subgraph $H$ of a graph $G$ is isometric if the distances between vertices in $H$ coincide with the distances between the corresponding vertices in $G$. We show that for any integer $n\ge 1$, there is a graph on $3^{n+O(\log^2 n)}$…

We present a series of almost settled inapproximability results for three fundamental problems. The first in our series is the subexponential-time inapproximability of the maximum independent set problem, a question studied in the area of…

We show that the problem of counting perfect matchings remains #P-complete even if we restrict the input to very dense graphs, proving the conjecture in [5]. Here "dense graphs" refer to bipartite graphs of bipartite independence number…

A graph $G$ is a $k$-prime product distance graph if its vertices can be labeled with distinct integers such that for any two adjacent vertices, the difference of their labels is the product of at most $k$ primes. A graph has prime product…

Let $k \geq 3$. We prove the following three bounds for the matching number, $\alpha'(G)$, of a graph, $G$, of order $n$ size $m$ and maximum degree at most $k$. If $k$ is odd, then $\alpha'(G) \ge \left( \frac{k-1}{k(k^2 - 3)} \right) n \,…

For a non-decreasing sequence $S=(s_1,s_2,\ldots)$ of positive integers, a partition of the vertex set of a graph $G$ into subsets $X_1,\ldots, X_\ell$, such that vertices in $X_i$ are pairwise at distance greater than $s_i$ for every…

We consider the problem of finding small distance-preserving subgraphs of undirected, unweighted interval graphs with $k$ terminal vertices. To start with, we show that finding an optimal distance-preserving subgraph is $\mathsf{NP}$-hard…

We initiate the study of the Diverse Pair of (Maximum/ Perfect) Matchings problems which given a graph $G$ and an integer $k$, ask whether $G$ has two (maximum/perfect) matchings whose symmetric difference is at least $k$. Diverse Pair of…

A mapping from the vertex set of a graph G = (V,E) into an interval of integers {0,...,k} is an L(2,1)-labelling of G of span k if any two adjacent vertices are mapped onto integers that are at least 2 apart, and every two vertices with a…

A $k$-L(2,1)-labeling of a graph is a function from its vertex set into the set $\{0,...,k\}$, such that the labels assigned to adjacent vertices differ by at least 2, and labels assigned to vertices of distance 2 are different. It is known…

Let $G = (V, E)$ be a graph without an isolated vertex. A set $D\subseteq V(G)$ is a $k$-distance paired domination set of $G$ if $D$ is a $k$-distance dominating set of $G$ and the induced subgraph $\langle D \rangle$ has a perfect…

The {\em packing chromatic number} $\chi_{\rho}(G)$ of a graph $G$ is the least integer $k$ for which there exists a mapping $f$ from $V(G)$ to $\{1,2,\ldots ,k\}$ such that any two vertices of color $i$ are at distance at least $i+1$. This…