• Corpus ID: 235790246

# Most Classic Problems Remain NP-hard on Relative Neighborhood Graphs and their Relatives

@article{Kunz2021MostCP,
title={Most Classic Problems Remain NP-hard on Relative Neighborhood Graphs and their Relatives},
author={Pascal Kunz and Till Fluschnik and Rolf Niedermeier and Malte Renken},
journal={ArXiv},
year={2021},
volume={abs/2107.04321}
}
Proximity graphs have been studied for several decades, motivated by applications in computational geometry, geography, data mining, and many other fields. However, the computational complexity of classic graph problems on proximity graphs mostly remained open. We now study 3-Colorability, Dominating Set, Feedback Vertex Set, Hamiltonian Cycle, and Independent Set on the proximity graph classes relative neighborhood graphs, Gabriel graphs, and relatively closest graphs. We prove that all of the…

## References

SHOWING 1-10 OF 45 REFERENCES
Some Simplified NP-Complete Graph Problems
• Computer Science, Mathematics
Theor. Comput. Sci.
• 1976
This paper shows that a number of NP - complete problems remain NP -complete even when their domains are substantially restricted, and determines essentially the lowest possible upper bounds on node degree for which the problems remainNP -complete.
Approximation algorithms for NP-complete problems on planar graphs
• B. Baker
• Mathematics, Computer Science
24th Annual Symposium on Foundations of Computer Science (sfcs 1983)
• 1983
A general technique that can be used to obtain approximation algorithms for various NP-complete problems on planar graphs, which includes maximum independent set, maximum tile salvage, partition into triangles, maximum H-matching, minimum vertex cover, minimum dominating set, and minimum edge dominating set.
PROXIMITY GRAPHS: E, δ, Δ, χ AND ω
• P. Bose, +6 authors D. Wood
• Mathematics, Computer Science
Int. J. Comput. Geom. Appl.
• 2012
Graph-theoretic properties of certain proximity graphs defined on planar point sets are investigated and bounds on the same parameters of some of the higher order versions are given.
Fixed Parameter Algorithms for DOMINATING SET and Related Problems on Planar Graphs
• Mathematics, Computer Science
Algorithmica
• 2002
An algorithm is presented that constructively produces a solution to the k -DOMINATING SET problem for planar graphs in time O(c^ \sqrt k n) where c=4^ 6\sqrt 34 and k is the size of the face cover set.
Coloring certain proximity graphs
Abstract We examine the graph coloring problem for three families of Euclidean proximity graphs. Results include a linear-time 4-coloring algorithm for relative neighborhood graphs , a linear-time
Nearest Neighbour Graph Realizability is NP-hard
• Mathematics, Computer Science
LATIN
• 1995
Several ways to define nearest neighbour graphs are given and it is shown that the problem of determining whether a given combinatorial graph can be realized as such a nearest neighbour graph is NP-hard.
Empty region graphs
• Computer Science, Mathematics
Comput. Geom.
• 2009
It is shown that every monotone property has at least one corresponding tight region; the possibilities and limitations of this general model for constructing a graph from a point set are discussed.
Two-Page Book Embeddings of 4-Planar Graphs
• Computer Science, Mathematics
Algorithmica
• 2015
This paper affirmatively answers the question whether this result can be extended to a class of graphs of degree greater than three and proposes a quadratic-time algorithm based on the book embedding viewpoint of the problem.
On Disjoint Cycles
It is shown, that for each constant k ≥ 1, the following problems can be solved in O(n) time: given a graph G, determine whether G has k vertex disjoint cycles, determine how many edges G has, and determineWhether G has a feedback vertex set of size ≤ k.
On Parameterized Exponential Time Complexity
• Mathematics, Computer Science
TAMC
• 2009
In this paper, we show that special instances of parameterized NP-hard problems are as difficult as the general instances in terms of their subexponential time computability. For example, we show