arXiv subjects

Egon Wanke

Publications and source records attributed to Egon Wanke.

13 recordsLinked to original sources

arXiv2026

Geodetic sets for directed acyclic planar geodetic graphs

A set of vertices $S$ of a directed graph $G$ is geodetic if every vertex of $G$ lies on a shortest path from a vertex of $S$ to a vertex of $S$. A directed graph is geodetic if there is at most one shortest path from every vertex of $G$ to every vertex of $G$. We prove the NP-completeness of the following decision problem. Given a directed acyclic planar geodetic graph $G$ and an integer $k$, does $G$ have a geodetic set with at most $k$ vertices? This implies that the question of whether $G$ has a strong or a monitoring geodetic set with at most $k$ vertices is also NP-complete for directed acyclic planar geodetic graphs. Furthermore, we prove that the number of vertices in a minimum geodetic set and the number of vertices in a minimum edge geodetic set can be computed in linear time for directed acyclic series-parallel graphs.

math.CO↗

arXiv2023

The directed metric dimension of directed co-graphs

A vertex $w$ resolves two vertices $u$ and $v$ in a directed graph $G$ if the distance from $w$ to $u$ is different to the distance from $w$ to $v$. A set of vertices $R$ is a resolving set for a directed graph $G$ if for every pair of vertices $u, v$ which are not in $R$ there is at least one vertex in $R$ that resolves $u$ and $v$ in $G$. The directed metric dimension of a directed graph $G$ is the size of a minimum resolving set for $G$. The decision problem Directed Metric Dimension for a given directed graph $G$ and a given number $k$ is the question whether $G$ has a resolving set of size at most $k$. In this paper, we study directed co-graphs. We introduce a linear time algorithm for computing a minimum resolving set for directed co-graphs and show that Directed Metric Dimension already is NP-complete for directed acyclic graphs.

cs.CC↗

arXiv2022

On the strong metric dimension of composed graphs

Two vertices $u$ and $v$ of an undirected graph $G$ are strongly resolved by a vertex $w$ if there is a shortest path between $w$ and $u$ containing $v$ or a shortest path between $w$ and $v$ containing $u$. A vertex set $R$ is a strong resolving set for $G$ if for each pair of vertices there is a vertex in $R$ that strongly resolves them. The strong metric dimension of $G$ is the size of a minimum strong resolving set for $G$. We show that a minimum strong resolving set for an undirected graph $G$ can be computed efficiently if and only if a minimum strong resolving set for each biconnected component of $G$ can be computed efficiently.

cs.CC↗

arXiv2021

On the Strong Metric Dimension of directed co-graphs

Let $G$ be a strongly connected directed graph and $u,v,w\in V(G)$ be three vertices. Then $w$ strongly resolves $u$ to $v$ if there is a shortest $u$-$w$-path containing $v$ or a shortest $w$-$v$-path containing $u$. A set $R\subseteq V(G)$ of vertices is a strong resolving set for a directed graph $G$ if for every pair of vertices $u,v\in V(G)$ there is at least one vertex in $R$ that strongly resolves $u$ to $v$ and at least one vertex in $R$ that strongly resolves $v$ to $u$. The distances of the vertices of $G$ to and from the vertices of a strong resolving set $R$ uniquely define the connectivity structure of the graph. The Strong Metric Dimension of a directed graph $G$ is the size of a smallest strong resolving set for $G$. The decision problem Strong Metric Dimension is the question whether $G$ has a strong resolving set of size at most $r$, for a given directed graph $G$ and a given number $r$. In this paper we study undirected and directed co-graphs and introduce linear time algorithms for Strong Metric Dimension. These algorithms can also compute strong resolving sets for co-graphs in linear time.

cs.CC↗

arXiv2021

A note on the complexity of k-Metric Dimension

Two vertices $u, v \in V$ of an undirected connected graph $G=(V,E)$ are resolved by a vertex $w$ if the distance between $u$ and $w$ and the distance between $v$ and $w$ are different. A set $R \subseteq V$ of vertices is a $k$-resolving set for $G$ if for each pair of vertices $u, v \in V$ there are at least $k$ distinct vertices $w_1,\ldots,w_k \in R$ such that each of them resolves $u$ and $v$. The $k$-Metric Dimension of $G$ is the size of a smallest $k$-resolving set for $G$. The decision problem $k$-Metric Dimension is the question whether G has a $k$-resolving set of size at most $r$, for a given graph $G$ and a given number $r$. In this paper, we proof the NP-completeness of $k$-Metric Dimension for bipartite graphs and each $k \geq 2$.

cs.CC↗

arXiv2020

Computing Directed Steiner Path Covers

In this article we consider the Directed Steiner Path Cover problem on directed co-graphs. Given a directed graph G=(V,E) and a subset T of V of so-called terminal vertices, the problem is to find a minimum number of vertex-disjoint simple directed paths, which contain all terminal vertices and a minimum number of non-terminal vertices (Steiner vertices). The primary minimization criteria is the number of paths. We show how to compute in linear time a minimum Steiner path cover for directed co-graphs. This leads to a linear time computation of an optimal directed Steiner path on directed co-graphs, if it exists. Since the Steiner path problem generalizes the Hamiltonian path problem, our results imply the first linear time algorithm for the directed Hamiltonian path problem on directed co-graphs. We also give binary integer programs for the (directed) Hamiltonian path problem, for the (directed) Steiner path problem, and for the (directed) Steiner path cover problem. These integer programs can be used to minimize change-over times in pick-and-place machines used by companies in electronic industry.

cs.DM↗

arXiv2019

The Fault-Tolerant Metric Dimension of Cographs

A vertex set $U \subseteq V$ of an undirected graph $G=(V,E)$ is a \textit{resolving set} for $G$ if for every two distinct vertices $u,v \in V$ there is a vertex $w \in U$ such that the distance between $u$ and $w$ and the distance between $v$ and $w$ are different. A resolving set $U$ is {\em fault-tolerant} if for every vertex $u\in U$ set $U\setminus \{u\}$ is still a resolving set. {The \em (fault-tolerant) Metric Dimension} of $G$ is the size of a smallest (fault-tolerant) resolving set for $G$. The {\em weighted (fault-tolerant) Metric Dimension} for a given cost function $c: V \longrightarrow \mathbb{R}_+$ is the minimum weight of all (fault-tolerant) resolving sets. Deciding whether a given graph $G$ has (fault-tolerant) Metric Dimension at most $k$ for some integer $k$ is known to be NP-complete. The weighted fault-tolerant Metric Dimension problem has not been studied extensively so far. In this paper we show that the weighted fault-tolerant metric dimension problem can be solved in linear time on cographs.

cs.DS↗

arXiv2018

Computing the metric dimension by decomposing graphs into extended biconnected components

A vertex set $U \subseteq V$ of an undirected graph $G=(V,E)$ is a $\textit{resolving set}$ for $G$, if for every two distinct vertices $u,v \in V$ there is a vertex $w \in U$ such that the distances between $u$ and $w$ and the distance between $v$ and $w$ are different. The $\textit{Metric Dimension}$ of $G$ is the size of a smallest resolving set for $G$. Deciding whether a given graph $G$ has Metric Dimension at most $k$ for some integer $k$ is well-known to be NP-complete. Many research has been done to understand the complexity of this problem on restricted graph classes. In this paper, we decompose a graph into its so called $\textit{extended biconnected components}$ and present an efficient algorithm for computing the metric dimension for a class of graphs having a minimum resolving set with a bounded number of vertices in every extended biconnected component. Further we show that the decision problem METRIC DIMENSION remains NP-complete when the above limitation is extended to usual biconnected components.

cs.CC↗

arXiv2016

Integer Programming Models and Parameterized Algorithms for Controlling Palletizers

We study the combinatorial FIFO Stack-Up problem, where bins have to be stacked-up from conveyor belts onto pallets. Given k sequences of labeled bins and a positive integer p, the goal is to stack-up the bins by iteratively removing the first bin of one of the k sequences and put it onto a pallet located at one of p stack-up places. The FIFO Stack-Up problem asks whether there is some processing of the sequences of bins such that at most p stack-up places are used. In this paper we strengthen the hardness of the FIFO Stack-Up by considering practical cases and the distribution of the pallets onto the sequences. We introduce a digraph model for this problem, the so called decision graph, which allows us to give a breadth first search solution. Further we apply methods to solve hard problems to the FIFO Stack-Up problem. In order to evaluate our algorithms, we introduce a method to generate random, but realistic instances for the FIFO Stack-Up problem. Our experimental study of running times shows that the breadth first search solution on the decision graph combined with a cutting technique can be used to solve practical instances on several thousands of bins of the FIFO Stack-Up problem. Further we analyze two integer programming approaches implemented in CPLEX and GLPK. As expected CPLEX can solve the instances much faster than GLPK and our pallet solution approach is much better than the bin solution approach.

cs.DS↗

arXiv2016

Minimum Power Range Assignment for Symmetric Connectivity in Sensor Networks with two Power Levels

This paper examines the problem of assigning a transmission power to every node of a wireless sensor network. The goal is to minimize the total power consumption while ensuring that the resulting communication graph is connected. We focus on a restricted version of this Range Assignment (RA) problem in which there are two different power levels. We only consider symmetrical transmission links to allow easy integration with low level wireless protocols that typically require bidirectional communication between two neighboring nodes. We introduce a parameterized polynomial time approximation algorithm with a performance ratio arbitrarily close to $π^2/6$. Additionally, we give an almost linear time approximation algorithm with a tight quality bound of $7/4$.

cs.DS↗

arXiv2015

Complexity of the FIFO Stack-Up Problem

We study the combinatorial FIFO stack-up problem. In delivery industry, bins have to be stacked-up from conveyor belts onto pallets with respect to customer orders. Given k sequences q_1, ..., q_k of labeled bins and a positive integer p, the aim is to stack-up the bins by iteratively removing the first bin of one of the k sequences and put it onto an initially empty pallet of unbounded capacity located at one of p stack-up places. Bins with different pallet labels have to be placed on different pallets, bins with the same pallet label have to be placed on the same pallet. After all bins for a pallet have been removed from the given sequences, the corresponding stack-up place will be cleared and becomes available for a further pallet. The FIFO stack-up problem is to find a stack-up sequence such that all pallets can be build-up with the available p stack-up places. In this paper, we introduce two digraph models for the FIFO stack-up problem, namely the processing graph and the sequence graph. We show that there is a processing of some list of sequences with at most p stack-up places if and only if the sequence graph of this list has directed pathwidth at most p-1. This connection implies that the FIFO stack-up problem is NP-complete in general, even if there are at most 6 bins for every pallet and that the problem can be solved in polynomial time, if the number p of stack-up places is assumed to be fixed. Further the processing graph allows us to show that the problem can be solved in polynomial time, if the number k of sequences is assumed to be fixed.

cs.DS↗

arXiv2013

Hierarchical Bipartition Routing for delivery guarantee in sparse wireless ad hoc sensor networks with obstacles

We introduce and evaluate a very simple landmark-based network partition technique called Hierarchical Bipartition Routing (HBR) to support routing with delivery guarantee in wireless ad hoc sensor networks. It is a simple routing protocol that can easily be combined with any other greedy routing algorithm to obtain delivery guarantee. The efficiency of HBR increases if the network is sparse and contains obstacles. The space necessary to store the additional routing information at a node u is on average not larger than the size necessary to store the IDs of the neighbors of u. The amount of work to setup the complete data structure is on average proportional to flooding the entire network log(n) times, where n is the total number of sensor nodes. We evaluate the performance of HBR in combination with two simple energy-aware geographic greedy routing algorithms based on physical coordinates and virtual coordinates, respectively. Our simulations show that the difference between using HBR and a weighted shortest path to escape a dead-end is only a few percent in typical cases.

cs.NI↗

arXiv2013

Metric Dimension for Gabriel Unit Disk Graphs is NP-Complete

We show that finding a minimal number of landmark nodes for a unique virtual addressing by hop-distances in wireless ad-hoc sensor networks is NP-complete even if the networks are unit disk graphs that contain only Gabriel edges. This problem is equivalent to Metric Dimension for Gabriel unit disk graphs. The Gabriel edges of a unit disc graph induce a planar O(\sqrt{n}) distance and an optimal energy spanner. This is one of the most interesting restrictions of Metric Dimension in the context of wireless multi-hop networks.

cs.CC↗