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 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 $3$.

CONNECTED LIAR'S DOMINATION IN GRAPHS: COMPLEXITY AND ALGORITHMS

2013 ◽  
Vol 05 (04) ◽  
pp. 1350024 ◽  
Author(s):  
B. S. PANDA ◽  
S. PAUL
Keyword(s):  
Dominating Set ◽  
Chordal Graphs ◽  
Hardness Of Approximation ◽  
Induced Subgraph ◽  
Path Graph ◽  
Block Graphs ◽  
Domination Problem ◽  
Np Complete ◽  
Domination In Graphs ◽  
General Graphs

A subset L ⊆ V of a graph G = (V, E) is called a connected liar's dominating set of G if (i) for all v ∈ V, |NG[v] ∩ L| ≥ 2, (ii) for every pair u, v ∈ V of distinct vertices, |(NG[u]∪NG[v])∩L| ≥ 3, and (iii) the induced subgraph of G on L is connected. In this paper, we initiate the algorithmic study of minimum connected liar's domination problem by showing that the corresponding decision version of the problem is NP-complete for general graph. Next we study this problem in subclasses of chordal graphs where we strengthen the NP-completeness of this problem for undirected path graph and prove that this problem is linearly solvable for block graphs. Finally, we propose an approximation algorithm for minimum connected liar's domination problem and investigate its hardness of approximation in general graphs.

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.

On the dominator coloring in proper interval graphs and block graphs

2015 ◽  
Vol 07 (04) ◽  
pp. 1550043 ◽  
Author(s):  
B. S. Panda ◽  
Arti Pandey
Keyword(s):  
Chromatic Number ◽  
Dominating Set ◽  
Domination Number ◽  
Maximum Size ◽  
Interval Graphs ◽  
Proper Coloring ◽  
Minimum Cardinality ◽  
Block Graphs ◽  
Minimum Number ◽  
Dominator Coloring

In a graph [Formula: see text], a vertex [Formula: see text] dominates a vertex [Formula: see text] if either [Formula: see text] or [Formula: see text] is adjacent to [Formula: see text]. A subset of vertex set [Formula: see text] that dominates all the vertices of [Formula: see text] is called a dominating set of graph [Formula: see text]. The minimum cardinality of a dominating set of [Formula: see text] is called the domination number of [Formula: see text] and is denoted by [Formula: see text]. A proper coloring of a graph [Formula: see text] is an assignment of colors to the vertices of [Formula: see text] such that any two adjacent vertices get different colors. The minimum number of colors required for a proper coloring of [Formula: see text] is called the chromatic number of [Formula: see text] and is denoted by [Formula: see text]. A dominator coloring of a graph [Formula: see text] is a proper coloring of the vertices of [Formula: see text] such that every vertex dominates all the vertices of at least one color class. The minimum number of colors required for a dominator coloring of [Formula: see text] is called the dominator chromatic number of [Formula: see text] and is denoted by [Formula: see text]. In this paper, we study the dominator chromatic number for the proper interval graphs and block graphs. We show that every proper interval graph [Formula: see text] satisfies [Formula: see text], and these bounds are sharp. For a block graph [Formula: see text], where one of the end block is of maximum size, we show that [Formula: see text]. We also characterize the block graphs with an end block of maximum size and attaining the lower bound.

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.

Pitchfork domination in graphs

2020 ◽  
Vol 12 (02) ◽  
pp. 2050025
Author(s):  
Manal N. Al-Harere ◽  
Mohammed A. Abdlhusein
Keyword(s):  
Maximum Degree ◽  
Undirected Graph ◽  
Minimum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
New Model ◽  
Domination In Graphs

In this paper, a new model of domination in graphs called the pitchfork domination is introduced. Let [Formula: see text] be a finite, simple and undirected graph without isolated vertices, a subset [Formula: see text] of [Formula: see text] is a pitchfork dominating set if every vertex [Formula: see text] dominates at least [Formula: see text] and at most [Formula: see text] vertices of [Formula: see text], where [Formula: see text] and [Formula: see text] are non-negative integers. The domination number of [Formula: see text], denotes [Formula: see text] is a minimum cardinality over all pitchfork dominating sets in [Formula: see text]. In this work, pitchfork domination when [Formula: see text] and [Formula: see text] is studied. Some bounds on [Formula: see text] related to the order, size, minimum degree, maximum degree of a graph and some properties are given. Pitchfork domination is determined for some known and new modified graphs. Finally, a question has been answered and discussed that; does every finite, simple and undirected graph [Formula: see text] without isolated vertices have a pitchfork domination or not?

scholarly journals On the Total Outer k-Independent Domination Number of Graphs

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 194 ◽  
Author(s):  
Abel Cabrera-Martínez ◽  
Juan Carlos Hernández-Gómez ◽  
Ernesto Parra-Inza ◽  
José María Sigarreta Almira
Keyword(s):  
Maximum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
Total Dominating Set ◽  
Np Hard Problem ◽  
Independent Domination ◽  
Independent Dominating Set ◽  
Computational Properties

A set of vertices of a graph G is a total dominating set if every vertex of G is adjacent to at least one vertex in such a set. We say that a total dominating set D is a total outer k-independent dominating set of G if the maximum degree of the subgraph induced by the vertices that are not in D is less or equal to k − 1 . The minimum cardinality among all total outer k-independent dominating sets is the total outer k-independent domination number of G. In this article, we introduce this parameter and begin with the study of its combinatorial and computational properties. For instance, we give several closed relationships between this novel parameter and other ones related to domination and independence in graphs. In addition, we give several Nordhaus–Gaddum type results. Finally, we prove that computing the total outer k-independent domination number of a graph G is an NP-hard problem.

On the Complexity of Reverse Minus and Signed Domination on Graphs

2015 ◽  
Vol 15 (01n02) ◽  
pp. 1550008
Author(s):  
CHUAN-MIN LEE ◽  
CHENG-CHIEN LO
Keyword(s):  
Planar Graphs ◽  
Decision Problem ◽  
Linear Time ◽  
Point Of View ◽  
Chordal Graphs ◽  
Fixed Parameter Tractable ◽  
Signed Domination ◽  
Domination Problem ◽  
Np Complete ◽  
Domination Problems

Motivated by the concept of reverse signed domination, we introduce the reverse minus domination problem on graphs, and study the reverse minus and signed domination problems from the algorithmic point of view. In this paper, we show that both the reverse minus and signed domination problems are polynomial-time solvable for strongly chordal graphs and distance-hereditary graphs, and are linear-time solvable for trees. For chordal graphs and bipartite planar graphs, however, we show that the decision problem corresponding to the reverse minus domination problem is NP-complete. For doubly chordal graphs and bipartite planar graphs, we show that the decision problem corresponding to the reverse signed domination problem is NP-complete. Furthermore, we show that even when restricted to bipartite planar graphs or doubly chordal graphs, the reverse signed domination problem is not fixed parameter tractable.

A UNIFIED APPROACH TO PARALLEL ALGORITHMS FOR THE DOMATIC PARTITION PROBLEM ON SPECIAL CLASSES OF PERFECT GRAPHS

1993 ◽  
Vol 03 (03) ◽  
pp. 233-241
Author(s):  
A. RAMAN ◽  
C. PANDU RANGAN
Keyword(s):  
Parallel Algorithms ◽  
Dominating Set ◽  
Interval Graphs ◽  
Perfect Graphs ◽  
Partition Problem ◽  
Dominating Sets ◽  
Unified Approach ◽  
Block Graphs ◽  
Vertex Set ◽  
Special Classes

A set of vertices D is a dominating set of a graph G=(V, E) if every vertex in V−D is adjacent to at least one vertex in D. The domatic partition of G is the partition of the vertex set V into a maximum number of dominating sets. In this paper, we present efficient parallel algorithms for finding the domatic partition of Interval graphs, Block graphs and K-trees.

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