Arne Leitert

Efficient Dominating and Edge Dominating Sets for Graphs and Hypergraphs

Let G = (V,E) be a graph. A vertex dominates itself and all its neighbors, i.e., every vertex v ∈ V dominates its closed neighborhood N[v]. A vertex set D in G is an efficient dominating (e.d.) set for G if for every vertex v ∈ V, there is exactly one d ∈ D dominating v. An edge set M ⊆ E is an efficient edge dominating (e.e.d.) set for G if it is an efficient dominating set in the line graph L(G) of G. The ED problem (EED problem, respectively) asks for the existence of an e.d. set (e.e.d. set, respectively) in the given graph.

Polynomial-time algorithms for weighted efficient domination problems in AT-free graphs and dually chordal graphs

An efficient dominating set (or perfect code) in a graph is a set of vertices the closed neighborhoods of which partition the vertex set of the graph. The minimum weight efficient domination problem is the problem of finding an efficient dominating set of minimum weight in a given vertex-weighted graph; the maximum weight efficient domination problem is defined similarly. We develop a framework for solving the weighted efficient domination problems based on a reduction to the maximum weight independent set problem in the square of the input graph. Using this approach, we improve on several previous results from the literature by deriving polynomial-time algorithms for the weighted efficient domination problems in the classes of dually chordal and AT-free graphs. In particular, this answers a question by Lu and Tang regarding the complexity of the minimum weight efficient domination problem in strongly chordal graphs. A framework for solving the minimum weighted efficient domination (Min-WED) problem is developed.The problem is reduced to the maximum weight independent set problem in the square.The Min-WED problem is polynomial for AT-free graphs and dually chordal graphs.In particular, this answers a question by Lu and Tang.The approach also works for the maximization version of the problem.

On the minimum eccentricity shortest path problem

Abstract In this paper, we introduce and investigate the Minimum Eccentricity Shortest Path (MESP) problem in unweighted graphs. It asks for a given graph to find a shortest path with minimum eccentricity. Let n and m denote the number of vertices and the number of edges of a given graph. We demonstrate that: • a minimum eccentricity shortest path plays a crucial role in obtaining the best to date approximation algorithm for a minimum distortion embedding of a graph into the line; • the MESP problem is NP-hard for planar bipartite graphs with maximum degree 3 and W[2]-hard for general graphs; • a shortest path of minimum eccentricity k can be computed in O ( n 2 k + 2 m ) time; • a 2-approximation, a 3-approximation, and an 8-approximation for the MESP problem can be computed in O ( n 3 ) time, in O ( n m ) time, and in O ( m ) time, respectively; • in a graph with a shortest path of eccentricity k, a k-dominating set can be found in n O ( k ) time.

On the Minimum Eccentricity Shortest Path Problem

In this paper, we introduce and investigate the Minimum Eccentricity Shortest Path (MESP) problem in unweighted graphs. It asks for a given graph to find a shortest path with minimum eccentricity. We demonstrate that: a minimum eccentricity shortest path plays a crucial role in obtaining the best to date approximation algorithm for a minimum distortion embedding of a graph into the line; the MESP-problem is NP-hard on general graphs; a 2-approximation, a 3-approximation, and an 8-approximation for the MESP-problem can be computed in \(\mathcal {O}(n^3)\) time, in \(\mathcal {O}(nm)\) time, and in linear time, respectively; a shortest path of minimum eccentricity k in general graphs can be computed in \(\mathcal {O}(n^{2k+2}m)\) time; the MESP-problem can be solved in linear time for trees.

Parameterized Approximation Algorithms for Some Location Problems in Graphs

We develop efficient parameterized, with additive error, approximation algorithms for the (Connected)

On Strong Tree-Breadth

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Parameterized approximation algorithms for some location problems in graphs

-Domination problem and the (Connected)

3-colouring for dually chordal graphs and generalisations

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Minimum Eccentricity Shortest Paths in Some Structured Graph Classes

-Center problem for unweighted and undirected graphs. Given a graph

Line-Distortion, Bandwidth and Path-Length of a Graph

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