86 research outputs found
Hitting Diamonds and Growing Cacti
We consider the following NP-hard problem: in a weighted graph, find a
minimum cost set of vertices whose removal leaves a graph in which no two
cycles share an edge. We obtain a constant-factor approximation algorithm,
based on the primal-dual method. Moreover, we show that the integrality gap of
the natural LP relaxation of the problem is \Theta(\log n), where n denotes the
number of vertices in the graph.Comment: v2: several minor changes
Revisiting path-type covering and partitioning problems
This is a survey article which is at the initial stage. The author will appreciate to receive your comments and contributions to improve the quality of the article. The author's contact address is [email protected] problems belong to the foundation of graph theory. There are several types of covering problems in graph theory such as covering the vertex set by stars (domination problem), covering the vertex set by cliques (clique covering problem), covering the vertex set by independent sets (coloring problem), and covering the vertex set by paths or cycles. A similar concept which is partitioning problem is also equally important. Lately research in graph theory has produced unprecedented growth because of its various application in engineering and science. The covering and partitioning problem by paths itself have produced a sizable volume of literatures. The research on these problems is expanding in multiple directions and the volume of research papers is exploding. It is the time to simplify and unify the literature on different types of the covering and partitioning problems. The problems considered in this article are path cover problem, induced path cover problem, isometric path cover problem, path partition problem, induced path partition problem and isometric path partition problem. The objective of this article is to summarize the recent developments on these problems, classify their literatures and correlate the inter-relationship among the related concepts
On the Computational Complexity of the Strong Geodetic Recognition Problem
A strong geodetic set of a graph~ is a vertex set~
in which it is possible to cover all the remaining vertices of~ by assigning a unique shortest path between each vertex pair of~. In the
Strong Geodetic problem (SG) a graph~ and a positive integer~ are given
as input and one has to decide whether~ has a strong geodetic set of
cardinality at most~. This problem is known to be NP-hard for general
graphs. In this work we introduce the Strong Geodetic Recognition problem
(SGR), which consists in determining whether even a given vertex set~ is strong geodetic. We demonstrate that this version is
NP-complete. We investigate and compare the computational complexity of both
decision problems restricted to some graph classes, deriving polynomial-time
algorithms, NP-completeness proofs, and initial parameterized complexity
results, including an answer to an open question in the literature for the
complexity of SG for chordal graphs
Generating partitions of a graph into a fixed number of minimum weight cuts
AbstractIn this paper, we present an algorithm for the generation of all partitions of a graph G with positive edge weights into k mincuts. The algorithm is an enumeration procedure based on the cactus representation of the mincuts of G. We report computational results demonstrating the efficiency of the algorithm in practice and describe in more detail a specific application for generating cuts in branch-and-cut algorithms for the traveling salesman problem
On Split-Coloring Problems
We study a new coloring concept which generalizes the classical vertex coloring problem in a graph by extending the notion of stable sets to split graphs. First of all, we propose the packing problem of finding the split graph of maximum size where a split graph is a graph G = (V,E) in which the vertex set V can be partitioned into a clique K and a stable set S. No condition is imposed on the edges linking vertices in S to the vertices in K. This maximum split graph problem gives rise to an associated partitioning problem that we call the split-coloring problem. Given a graph, the objective is to cover all his vertices by a least number of split graphs. Definitions related to this new problem are introduced. We mention some polynomially solvable cases and describe open questions on this are
Network Sparsification for Steiner Problems on Planar and Bounded-Genus Graphs
We propose polynomial-time algorithms that sparsify planar and bounded-genus
graphs while preserving optimal or near-optimal solutions to Steiner problems.
Our main contribution is a polynomial-time algorithm that, given an unweighted
graph embedded on a surface of genus and a designated face bounded
by a simple cycle of length , uncovers a set of size
polynomial in and that contains an optimal Steiner tree for any set of
terminals that is a subset of the vertices of .
We apply this general theorem to prove that: * given an unweighted graph
embedded on a surface of genus and a terminal set , one
can in polynomial time find a set that contains an optimal
Steiner tree for and that has size polynomial in and ; * an
analogous result holds for an optimal Steiner forest for a set of terminal
pairs; * given an unweighted planar graph and a terminal set , one can in polynomial time find a set that contains
an optimal (edge) multiway cut separating and that has size polynomial
in .
In the language of parameterized complexity, these results imply the first
polynomial kernels for Steiner Tree and Steiner Forest on planar and
bounded-genus graphs (parameterized by the size of the tree and forest,
respectively) and for (Edge) Multiway Cut on planar graphs (parameterized by
the size of the cutset). Additionally, we obtain a weighted variant of our main
contribution
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