arXiv subjects

A. Shakiba

Publications and source records attributed to A. Shakiba.

6 recordsLinked to original sources

arXiv2018

On fixed-parameter tractability of the mixed domination problem for graphs with bounded tree-width

A mixed dominating set for a graph $G = (V,E)$ is a set $S\subseteq V \cup E$ such that every element $x \in (V \cup E) \backslash S$ is either adjacent or incident to an element of $S$. The mixed domination number of a graph $G$, denoted by $γ_m(G)$, is the minimum cardinality of mixed dominating sets of $G$. Any mixed dominating set with the cardinality of $γ_m(G)$ is called a minimum mixed dominating set. The mixed domination set (MDS) problem is to find a minimum mixed dominating set for a graph $G$ and is known to be an NP-complete problem. In this paper, we present a novel approach to find all of the mixed dominating sets, called the AMDS problem, of a graph with bounded tree-width $tw$. Our new technique of assigning power values to edges and vertices, and combining with dynamic programming, leads to a fixed-parameter algorithm of time $O(3^{tw^{2}}\times tw^2 \times |V|)$. This shows that MDS is fixed-parameter tractable with respect to tree-width. In addition, we theoretically improve the proposed algorithm to solve the MDS problem in $O(6^{tw} \times |V|)$ time.

cs.DM↗

arXiv2018

Domination Cover Number of Graphs

A set $D \subseteq V$ for the graph $G=(V, E)$ is called a dominating set if any vertex $v\in V\setminus D$ has at least one neighbor in $D$. Fomin et al.[9] gave an algorithm for enumerating all minimal dominating sets with $n$ vertices in $O(1.7159^n)$ time. It is known that the number of minimal dominating sets for interval graphs and trees on $n$ vertices is at most $3^{n/3} \approx 1.4422^n$. In this paper, we introduce the domination cover number as a new criterion for evaluating the dominating sets in graphs. The domination cover number of a dominating set $D$, denoted by $\mathcal{C}_D(G)$, is the summation of the degrees of the vertices in $D$. Maximizing or minimizing this parameter among all minimal dominating sets have interesting applications in many real-world problems, such as the art gallery problem. Moreover, we investigate this concept for different graph classes and propose some algorithms for finding the domination cover number in trees, block graphs.

cs.DM↗

arXiv2018

On the complexity of the outer-connected bondage and the outer-connected reinforcement problems

Let $G=(V,E)$ be a graph. A subset $S \subseteq V$ is a dominating set of $G$ if every vertex not in $S$ is adjacent to a vertex in $S$. A set $\tilde{D} \subseteq V$ of a graph $G=(V,E) $ is called an outer-connected dominating set for $G$ if (1) $\tilde{D}$ is a dominating set for $G$, and (2) $G [V \setminus \tilde{D}]$, the induced subgraph of $G$ by $V \setminus \tilde{D}$, is connected. The minimum size among all outer-connected dominating sets of $G$ is called the outer-connected domination number of $G$ and is denoted by $\tildeγ_c(G)$. We define the outer-connected bondage number of a graph $G$ as the minimum number of edges whose removal from $G$ results in a graph with an outer-connected domination number larger than the one for $G$. Also, the outer-connected reinforcement number of a graph $G$ is defined as the minimum number of edges whose addition to $G$ results in a graph with an outer-connected domination number, which is smaller than the one for $G$. This paper shows that the decision problems for the outer-connected bondage and the outer-connected reinforcement numbers are $\mathbf{NP}$-hard. Also, the exact values of the bondage number are determined for several classes of graphs.

cs.DM↗

arXiv2017

On outer-connected domination for graph products

An outer-connected dominating set for an arbitrary graph $G$ is a set $\tilde{D} \subseteq V$ such that $\tilde{D}$ is a dominating set and the induced subgraph $G [V \setminus \tilde{D}]$ be connected. In this paper, we focus on the outer-connected domination number of the product of graphs. We investigate the existence of outer-connected dominating set in lexicographic product and Corona of two arbitrary graphs, and we present upper bounds for outer-connected domination number in lexicographic and Cartesian product of graphs. Also, we establish an equivalent form of the Vizing's conjecture for outer-connected domination number in lexicographic and Cartesian product as $\tilde{γ_c}(G \circ K)\tilde{γ_c}(H \circ K) \leq \tilde{γ_c}(G\Box H)\circ K$. Furthermore, we study the outer-connected domination number of the direct product of finitely many complete graphs.

cs.DM↗

arXiv2017

An Efficient Algorithm for Mixed Domination on Generalized Series-Parallel Graphs

A mixed dominating set $S$ of a graph $G=(V,E)$ is a subset $ S \subseteq V \cup E$ such that each element $v\in (V \cup E) \setminus S$ is adjacent or incident to at least one element in $S$. The mixed domination number $γ_m(G)$ of a graph $G$ is the minimum cardinality among all mixed dominating sets in $G$. The problem of finding $γ_{m}(G)$ is know to be NP-complete. In this paper, we present an explicit polynomial-time algorithm to construct a mixed dominating set of size $γ_{m}(G)$ by a parse tree when $G$ is a generalized series-parallel graph.

cs.DM↗

arXiv2016

A combinatorial approach to certain topological spaces based on minimum complement S-approximation spaces

An S-approximation space is a novel approach to study systems with uncertainty that are not expressible in terms of inclusion relations. In this work, we further examined these spaces, mostly from a topological point of view by a combinatorial approach. This work also identifies a subclass of these approximation spaces, called $S_\mathcal{MC}$-approximations. Topological properties of this subclass are investigated and finally, the topologies formed by $S_\mathcal{MC}$-approximations are enumerated up to homeomorphism.

math.AT↗