arXiv subjects

Didem Gözüpek

Publications and source records attributed to Didem Gözüpek.

24 records · Page 2Linked to original sources

arXiv2017

Characterizations of minimal dominating sets and the well-dominated property in lexicographic product graphs

A graph is said to be well-dominated if all its minimal dominating sets are of the same size. The class of well-dominated graphs forms a subclass of the well studied class of well-covered graphs. While the recognition problem for the class of well-covered graphs is known to be co-NP-complete, the recognition complexity of well-dominated graphs is open. In this paper we introduce the notion of an irreducible dominating set, a variant of dominating set generalizing both minimal dominating sets and minimal total dominating sets. Based on this notion, we characterize the family of minimal dominating sets in a lexicographic product of two graphs and derive a characterization of the well-dominated lexicographic product graphs. As a side result motivated by this study, we give a polynomially testable characterization of well-dominated graphs with domination number two, and show, more generally, that well-dominated graphs can be recognized in polynomial time in any class of graphs with bounded domination number. Our results include a characterization of dominating sets in lexicographic product graphs, which generalizes the expression for the domination number of such graphs following from works of Zhang et al. (2011) and of Šumenjak et al. (2012).

cs.DM↗

arXiv2017

Parameterized complexity of finding a spanning tree with minimum reload cost diameter

We study the minimum diameter spanning tree problem under the reload cost model (DIAMETER-TREE for short) introduced by Wirth and Steffan (2001). In this problem, given an undirected edge-colored graph $G$, reload costs on a path arise at a node where the path uses consecutive edges of different colors. The objective is to find a spanning tree of $G$ of minimum diameter with respect to the reload costs. We initiate a systematic study of the parameterized complexity of the DIAMETER-TREE problem by considering the following parameters: the cost of a solution, and the treewidth and the maximum degree $Δ$ of the input graph. We prove that DIAMETER-TREE is para-NP-hard for any combination of two of these three parameters, and that it is FPT parameterized by the three of them. We also prove that the problem can be solved in polynomial time on cactus graphs. This result is somehow surprising since we prove DIAMETER-TREE to be NP-hard on graphs of treewidth two, which is best possible as the problem can be trivially solved on forests. When the reload costs satisfy the triangle inequality, Wirth and Steffan (2001) proved that the problem can be solved in polynomial time on graphs with $Δ= 3$, and Galbiati (2008) proved that it is NP-hard if $Δ= 4$. Our results show, in particular, that without the requirement of the triangle inequality, the problem is NP-hard if $Δ= 3$, which is also best possible. Finally, in the case where the reload costs are polynomially bounded by the size of the input graph, we prove that DIAMETER-TREE is in XP and W[1]-hard parameterized by the treewidth plus $Δ$.

cs.DS↗

arXiv2016

Edge Coloring with Minimum Reload/Changeover Costs

In an edge-colored graph, a traversal cost occurs at a vertex along a path when consecutive edges with different colors are traversed. The value of the traversal cost depends only on the colors of the traversed edges. This concept leads to two global cost measures, namely the \emph{reload cost} and the \emph{changeover cost}, that have been studied in the literature and have various applications in telecommunications, transportation networks, and energy distribution networks. Previous work focused on problems with an edge-colored graph being part of the input. In this paper, we formulate and focus on two pairs of problems that aim to find an edge coloring of a graph so as to minimize the reload and changeover costs. The first pair of problems aims to find a proper edge coloring so that the reload/changeover cost of a set of paths is minimized. The second pair of problems aim to find a proper edge coloring and a spanning tree so that the reload/changeover cost is minimized. We present several hardness results as well as polynomial-time solvable special cases.

cs.DS↗

arXiv2016

A Simple Extension of Dirac's Theorem on Hamiltonicity

The classical Dirac theorem asserts that every graph $G$ on $n$ vertices with minimum degree $δ(G) \ge \lceil n/2 \rceil$ is Hamiltonian. The lower bound of $\lceil n/2 \rceil$ on the minimum degree of a graph is tight. In this paper, we extend the classical Dirac theorem to the case where $δ(G) \ge \lfloor n/2 \rfloor $ by identifying the only non-Hamiltonian graph families in this case. We first present a short and simple proof. We then provide an alternative proof that is constructive and self-contained. Consequently, we provide a polynomial-time algorithm that constructs a Hamiltonian cycle, if exists, of a graph $G$ with $δ(G) \ge \lfloor n/2 \rfloor$, or determines that the graph is non-Hamiltonian. Finally, we present a self-contained proof for our algorithm which provides insight into the structure of Hamiltonian cycles when $δ(G) \ge \lfloor n/2 \rfloor$ and is promising for extending the results of this paper to the cases with smaller degree bounds.

cs.DM↗

arXiv2016

Parameterized complexity of the MINCCA problem on graphs of bounded decomposability

In an edge-colored graph, the cost incurred at a vertex on a path when two incident edges with different colors are traversed is called reload or changeover cost. The "Minimum Changeover Cost Arborescence" (MINCCA) problem consists in finding an arborescence with a given root vertex such that the total changeover cost of the internal vertices is minimized. It has been recently proved by Gözüpek et al. [TCS 2016] that the problem is FPT when parameterized by the treewidth and the maximum degree of the input graph. In this article we present the following results for the MINCCA problem: - the problem is W[1]-hard parameterized by the treedepth of the input graph, even on graphs of average degree at most 8. In particular, it is W[1]-hard parameterized by the treewidth of the input graph, which answers the main open problem of Gözüpek et al. [TCS 2016]; - it is W[1]-hard on multigraphs parameterized by the tree-cutwidth of the input multigraph; - it is FPT parameterized by the star tree-cutwidth of the input graph, which is a slightly restricted version of tree-cutwidth. This result strictly generalizes the FPT result given in Gözüpek et al. [TCS 2016]; - it remains NP-hard on planar graphs even when restricted to instances with at most 6 colors and 0/1 symmetric costs, or when restricted to instances with at most 8 colors, maximum degree bounded by 4, and 0/1 symmetric costs.

cs.DS↗

arXiv2013

A Fair Scheduling Model for Centralized Cognitive Radio Networks

We formulate throughput maximizing, max-min fair, weighted max-min fair, and proportionally fair scheduling problems for cognitive radio networks managed by a centralized cognitive base station. We propose a very general scheduling model accomplishing goals such as making frequency, time slot, and data rate allocation to secondary users with possibly multiple antennas, in a heterogenous multi-channel and multi-user scenario. Moreover, our schedulers ensure that reliable communication between the cognitive base station and secondary users are maintained, no collisions occur among secondary users, and primary users in the service area of the cognitive base station are not disturbed. Two distinctive features of our fair schedulers are that they provide joint temporal and throughput fairness, and take throughput values experienced by secondary users in the recent past, referred to as window size, into account and use this information in the current scheduling decision. We also propose a heuristic algorithm for our fair schedulers and demonstrate through simulations that our proposed heuristic yields very close solutions to the values obtained from the optimization softwares. Furthermore, we make extensive simulations to evaluate our schedulers' performance in terms of both total throughput and fairness for varying number of secondary users, frequencies, antennas, and window size.

cs.NI↗