scholarly journals Semipaired Domination in Some Subclasses of Chordal Graphs

2021 ◽  
Vol vol. 23 no. 1 (Discrete Algorithms) ◽  
Author(s):  
Michael A. Henning ◽  
Arti Pandey ◽  
Vikash Tripathi
Keyword(s):  
Linear Time ◽  
Dominating Set ◽  
Domination Number ◽  
Time Algorithm ◽  
Chordal Graphs ◽  
Minimum Cardinality ◽  
Positive Side ◽  
Domination Problem ◽  
Np Complete ◽  
Decision Version

A dominating set $D$ of a graph $G$ without isolated vertices is called semipaired dominating set if $D$ can be partitioned into $2$-element subsets such that the vertices in each set are at distance at most $2$. The semipaired domination number, denoted by $\gamma_{pr2}(G)$ is the minimum cardinality of a semipaired dominating set of $G$. Given a graph $G$ with no isolated vertices, the \textsc{Minimum Semipaired Domination} problem is to find a semipaired dominating set of $G$ of cardinality $\gamma_{pr2}(G)$. The decision version of the \textsc{Minimum Semipaired Domination} problem is already known to be NP-complete for chordal graphs, an important graph class. In this paper, we show that the decision version of the \textsc{Minimum Semipaired Domination} problem remains NP-complete for split graphs, a subclass of chordal graphs. On the positive side, we propose a linear-time algorithm to compute a minimum cardinality semipaired dominating set of block graphs. In addition, we prove that the \textsc{Minimum Semipaired Domination} problem is APX-complete for graphs with maximum degree $3$.

scholarly journals Complexity issues of perfect secure domination in graphs

2021 ◽  
Vol 55 ◽  
pp. 11
Author(s):  
P. Chakradhar ◽  
P. Venkata Subba Reddy
Keyword(s):  
Linear Time ◽  
Dominating Set ◽  
Domination Number ◽  
Time Algorithm ◽  
Bipartite Graphs ◽  
Chordal Graphs ◽  
Minimum Cardinality ◽  
Split Graphs ◽  
Secure Domination ◽  
Convex Bipartite Graphs

Let G = (V, E) be a simple, undirected and connected graph. A dominating set S is called a secure dominating set if for each u ∈ V \ S, there exists v ∈ S such that (u, v) ∈ E and (S \{v}) ∪{u} is a dominating set of G. If further the vertex v ∈ S is unique, then S is called a perfect secure dominating set (PSDS). The perfect secure domination number γps(G) is the minimum cardinality of a perfect secure dominating set of G. Given a graph G and a positive integer k, the perfect secure domination (PSDOM) problem is to check whether G has a PSDS of size at most k. In this paper, we prove that PSDOM problem is NP-complete for split graphs, star convex bipartite graphs, comb convex bipartite graphs, planar graphs and dually chordal graphs. We propose a linear time algorithm to solve the PSDOM problem in caterpillar trees and also show that this problem is linear time solvable for bounded tree-width graphs and threshold graphs, a subclass of split graphs. Finally, we show that the domination and perfect secure domination problems are not equivalent in computational complexity aspects.

The strong domination problem in block graphs and proper interval graphs

2019 ◽  
Vol 11 (06) ◽  
pp. 1950063
Author(s):  
Saikat Pal ◽  
D. Pradhan
Keyword(s):  
Maximum Degree ◽  
Dominating Set ◽  
Interval Graphs ◽  
Chordal Graphs ◽  
Minimum Cardinality ◽  
Positive Side ◽  
Degree Of A Vertex ◽  
Block Graphs ◽  
Domination Problem ◽  
Np Complete

In a graph [Formula: see text], the degree of a vertex [Formula: see text], denoted by [Formula: see text], is defined as the number of edges incident on [Formula: see text]. A set [Formula: see text] of vertices of [Formula: see text] is called a strong dominating set if for every [Formula: see text], there exists a vertex [Formula: see text] such that [Formula: see text] and [Formula: see text]. For a given graph [Formula: see text], Min-Strong-DS is the problem of finding a strong dominating set of minimum cardinality. The decision version of Min-Strong-DS is shown to be NP -complete for chordal graphs. In this paper, we present polynomial time algorithms for computing a strong dominating set in block graphs and proper interval graphs, two subclasses of chordal graphs. On the other hand, we show that for a graph [Formula: see text] with [Formula: see text]-vertices, Min-Strong-DS cannot be approximated within a factor of [Formula: see text] for every [Formula: see text], unless NP [Formula: see text] DTIME ([Formula: see text]). We also show that Min-Strong-DS is APX -complete for graphs with maximum degree [Formula: see text]. On the positive side, we show that Min-Strong-DS can be approximated within a factor of [Formula: see text] for graphs with maximum degree [Formula: see text].

scholarly journals A linear time algorithm for minimum equitable dominating set in trees

2021 ◽  
Vol 40 (4) ◽  
pp. 805-814
Author(s):  
Sohel Rana ◽  
Sk. Md. Abu Nayeem
Keyword(s):  
Linear Time ◽  
Dominating Set ◽  
Domination Number ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Minimum Cardinality ◽  
Np Complete ◽  
General Graphs

Let G = (V, E) be a graph. A subset De of V is said to be an equitable dominating set if for every v ∈ V \ De there exists u ∈ De such that uv ∈ E and |deg(u) − deg(v)| ≤ 1, where, deg(u) and deg(v) denote the degree of the vertices u and v respectively. An equitable dominating set with minimum cardinality is called the minimum equitable dominating set and its cardinality is called the equitable domination number and it is denoted by γe. The problem of finding minimum equitable dominating set in general graphs is NP-complete. In this paper, we give a linear time algorithm to determine minimum equitable dominating set of a tree.

scholarly journals Total Roman {3}-Domination: The Complexity and Linear-Time Algorithm for Trees

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 293
Author(s):  
Xinyue Liu ◽  
Huiqin Jiang ◽  
Pu Wu ◽  
Zehui Shao
Keyword(s):  
Linear Time ◽  
Minimum Weight ◽  
Domination Number ◽  
Time Algorithm ◽  
Simple Graph ◽  
Linear Time Algorithm ◽  
Domination Problem ◽  
Np Complete ◽  
Chordal Bipartite Graphs ◽  
Dominating Function

For a simple graph G=(V,E) with no isolated vertices, a total Roman {3}-dominating function(TR3DF) on G is a function f:V(G)→{0,1,2,3} having the property that (i) ∑w∈N(v)f(w)≥3 if f(v)=0; (ii) ∑w∈N(v)f(w)≥2 if f(v)=1; and (iii) every vertex v with f(v)≠0 has a neighbor u with f(u)≠0 for every vertex v∈V(G). The weight of a TR3DF f is the sum f(V)=∑v∈V(G)f(v) and the minimum weight of a total Roman {3}-dominating function on G is called the total Roman {3}-domination number denoted by γt{R3}(G). In this paper, we show that the total Roman {3}-domination problem is NP-complete for planar graphs and chordal bipartite graphs. Finally, we present a linear-time algorithm to compute the value of γt{R3} for trees.

k-Efficient domination: Algorithmic perspective

2022 ◽  
Author(s):  
Mohsen Alambardar Meybodi
Keyword(s):  
Decision Problem ◽  
Dominating Set ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Chordal Graphs ◽  
Sparse Graphs ◽  
Fpt Algorithm ◽  
Domination Problem ◽  
Efficient Dominating Set ◽  
Np Complete

A set [Formula: see text] of a graph [Formula: see text] is called an efficient dominating set of [Formula: see text] if every vertex [Formula: see text] has exactly one neighbor in [Formula: see text], in other words, the vertex set [Formula: see text] is partitioned to some circles with radius one such that the vertices in [Formula: see text] are the centers of partitions. A generalization of this concept, introduced by Chellali et al. [k-Efficient partitions of graphs, Commun. Comb. Optim. 4 (2019) 109–122], is called [Formula: see text]-efficient dominating set that briefly partitions the vertices of graph with different radiuses. It leads to a partition set [Formula: see text] such that each [Formula: see text] consists a center vertex [Formula: see text] and all the vertices in distance [Formula: see text], where [Formula: see text]. In other words, there exist the dominators with various dominating powers. The problem of finding minimum set [Formula: see text] is called the minimum [Formula: see text]-efficient domination problem. Given a positive integer [Formula: see text] and a graph [Formula: see text], the [Formula: see text]-efficient Domination Decision problem is to decide whether [Formula: see text] has a [Formula: see text]-efficient dominating set of cardinality at most [Formula: see text]. The [Formula: see text]-efficient Domination Decision problem is known to be NP-complete even for bipartite graphs [M. Chellali, T. W. Haynes and S. Hedetniemi, k-Efficient partitions of graphs, Commun. Comb. Optim. 4 (2019) 109–122]. Clearly, every graph has a [Formula: see text]-efficient dominating set but it is not correct for efficient dominating set. In this paper, we study the following: [Formula: see text]-efficient domination problem set is NP-complete even in chordal graphs. A polynomial-time algorithm for [Formula: see text]-efficient domination in trees. [Formula: see text]-efficient domination on sparse graphs from the parametrized complexity perspective. In particular, we show that it is [Formula: see text]-hard on d-degenerate graphs while the original dominating set has Fixed Parameter Tractable (FPT) algorithm on d-degenerate graphs. [Formula: see text]-efficient domination on nowhere-dense graphs is FPT.

Total vertex-edge domination in graphs: Complexity and algorithms

2021 ◽  
Author(s):  
Nitisha Singhwal ◽  
Palagiri Venkata Subba Reddy
Keyword(s):  
Linear Time ◽  
Dominating Set ◽  
Domination Number ◽  
Bipartite Graphs ◽  
Chordal Graphs ◽  
Induced Subgraph ◽  
Minimum Cardinality ◽  
Edge Dominating Set ◽  
Linear Programming Formulation ◽  
Convex Bipartite Graphs

Let [Formula: see text] be a simple, undirected and connected graph. A vertex [Formula: see text] of a simple, undirected graph [Formula: see text]-dominates all edges incident to at least one vertex in its closed neighborhood [Formula: see text]. A set [Formula: see text] of vertices is a vertex-edge dominating set of [Formula: see text], if every edge of graph [Formula: see text] is [Formula: see text]-dominated by some vertex of [Formula: see text]. A vertex-edge dominating set [Formula: see text] of [Formula: see text] is called a total vertex-edge dominating set if the induced subgraph [Formula: see text] has no isolated vertices. The total vertex-edge domination number [Formula: see text] is the minimum cardinality of a total vertex-edge dominating set of [Formula: see text]. In this paper, we prove that the decision problem corresponding to [Formula: see text] is NP-complete for chordal graphs, star convex bipartite graphs, comb convex bipartite graphs and planar graphs. The problem of determining [Formula: see text] of a graph [Formula: see text] is called the minimum total vertex-edge domination problem (MTVEDP). We prove that MTVEDP is linear time solvable for chain graphs and threshold graphs. We also show that MTVEDP can be approximated within approximation ratio of [Formula: see text]. It is shown that the domination and total vertex-edge domination problems are not equivalent in computational complexity aspects. Finally, an integer linear programming formulation for MTVEDP is presented.

A linear time algorithm to compute a minimum restrained dominating set in proper interval graphs

2015 ◽  
Vol 07 (02) ◽  
pp. 1550020 ◽  
Author(s):  
B. S. Panda ◽  
D. Pradhan
Keyword(s):  
Linear Time ◽  
Dominating Set ◽  
Time Algorithm ◽  
Interval Graph ◽  
Chordal Graphs ◽  
Linear Time Algorithm ◽  
Circle Graphs ◽  
Restrained Domination ◽  
Proper Interval Graph ◽  
Domination Problem

A set D ⊆ V is a restrained dominating set of a graph G = (V, E) if every vertex in V\D is adjacent to a vertex in D and a vertex in V\D. Given a graph G and a positive integer k, the restrained domination problem is to check whether G has a restrained dominating set of size at most k. The restrained domination problem is known to be NP-complete even for chordal graphs. In this paper, we propose a linear time algorithm to compute a minimum restrained dominating set of a proper interval graph. We present a polynomial time reduction that proves the NP-completeness of the restrained domination problem for undirected path graphs, chordal bipartite graphs, circle graphs, and planar graphs.

Algorithmic aspects of k-part degree restricted domination in graphs

2020 ◽  
Vol 12 (05) ◽  
pp. 2050057
Author(s):  
S. S. Kamath ◽  
A. Senthil Thilak ◽  
M. Rashmi
Keyword(s):  
Communication Networks ◽  
Polynomial Time ◽  
Dominating Set ◽  
Domination Number ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Bipartite Graphs ◽  
Chordal Graphs ◽  
Circle Graphs ◽  
Minimum Cardinality

The concept of network is predominantly used in several applications of computer communication networks. It is also a fact that the dominating set acts as a virtual backbone in a communication network. These networks are vulnerable to breakdown due to various causes, including traffic congestion. In such an environment, it is necessary to regulate the traffic so that these vulnerabilities could be reasonably controlled. Motivated by this, [Formula: see text]-part degree restricted domination is defined as follows. For a positive integer [Formula: see text], a dominating set [Formula: see text] of a graph [Formula: see text] is said to be a [Formula: see text]-part degree restricted dominating set ([Formula: see text]-DRD set) if for all [Formula: see text], there exists a set [Formula: see text] such that [Formula: see text] and [Formula: see text]. The minimum cardinality of a [Formula: see text]-DRD set of a graph [Formula: see text] is called the [Formula: see text]-part degree restricted domination number of [Formula: see text] and is denoted by [Formula: see text]. In this paper, we present a polynomial time reduction that proves the NP -completeness of the [Formula: see text]-part degree restricted domination problem for bipartite graphs, chordal graphs, undirected path graphs, chordal bipartite graphs, circle graphs, planar graphs and split graphs. We propose a polynomial time algorithm to compute a minimum [Formula: see text]-DRD set of a tree and minimal [Formula: see text]-DRD set of a graph.

scholarly journals Upper k-tuple domination in graphs

2012 ◽  
Vol Vol. 14 no. 2 (Graph Theory) ◽  
Author(s):  
Gerard Jennhwa Chang ◽  
Paul Dorbec ◽  
Hye Kyung Kim ◽  
André Raspaud ◽  
Haichao Wang ◽  
...  
Keyword(s):  
Upper Bound ◽  
Dominating Set ◽  
Domination Number ◽  
Chordal Graphs ◽  
Regular Graphs ◽  
International Audience ◽  
Algorithmic Aspect ◽  
Domination Problem ◽  
Np Complete ◽  
Domination In Graphs

Graph Theory International audience For a positive integer k, a k-tuple dominating set of a graph G is a subset S of V (G) such that |N [v] ∩ S| ≥ k for every vertex v, where N [v] = {v} ∪ {u ∈ V (G) : uv ∈ E(G)}. The upper k-tuple domination number of G, denoted by Γ×k (G), is the maximum cardinality of a minimal k-tuple dominating set of G. In this paper we present an upper bound on Γ×k (G) for r-regular graphs G with r ≥ k, and characterize extremal graphs achieving the upper bound. We also establish an upper bound on Γ×2 (G) for claw-free r-regular graphs. For the algorithmic aspect, we show that the upper k-tuple domination problem is NP-complete for bipartite graphs and for chordal graphs.

On double edge-domination and total domination of trees

2021 ◽  
pp. 1-8
Author(s):  
Bünyamin Şahin ◽  
Abdulgani Şahin
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Chordal Graphs ◽  
Total Domination ◽  
Total Domination Number ◽  
Minimum Cardinality ◽  
Double Edge ◽  
Np Complete

In a graph G, a vertex v is dominated by an edge e, if e is incident with v or e is incident with a vertex which is a neighbor of v. An edge-vertex dominating set D is a subset of the edge set of G such that every vertex of G is edge-vertex dominated by an edge of D. The ev-domination number equals to the number of an edge-vertex dominating set of G which has minimum cardinality and it is denoted by γev (G). We here analyze double edge-vertex domination such that a double edge-vertex dominating set D is a subset of the edge set of G, provided that all vertices in G are ev-dominated by at least two edges of D. The double ev-domination number equals to the number of an double edge-vertex dominating set of G which has minimum cardinality and it is denoted by γdev (G). We demonstrate that the enumeration of the double ev-domination number of chordal graphs is NP-complete. Moreover several results about total domination number and double ev-domination number are obtained for trees.

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