2-Point set domination in separable graphs

2021 ◽  
Author(s):  
Purnima Gupta ◽  
Deepti Jain
Keyword(s):  
Nonempty Subset ◽  
Dominating Set ◽  
Domination Number ◽  
Induced Subgraph ◽  
Minimum Cardinality ◽  
Point Set ◽  
Separable Graph

In a graph [Formula: see text], a set [Formula: see text] is a [Formula: see text]-point set dominating set (in short 2-psd set) of [Formula: see text] if for every subset [Formula: see text] there exists a nonempty subset [Formula: see text] containing at most two vertices such that the induced subgraph [Formula: see text] is connected in [Formula: see text]. The [Formula: see text]-point set domination number of [Formula: see text], denoted by [Formula: see text], is the minimum cardinality of a 2-psd set of [Formula: see text]. The main focus of this paper is to find the value of [Formula: see text] for a separable graph and thereafter computing [Formula: see text] for some well-known classes of separable graphs. Further we classify the set of all 2-psd sets of a separable graph into six disjoint classes and study the existence of minimum 2-psd sets in each class.

Bounds on 2-point set domination number of a graph

2018 ◽  
Vol 10 (01) ◽  
pp. 1850012
Author(s):  
Purnima Gupta ◽  
Deepti Jain
Keyword(s):  
Lower Bounds ◽  
Upper Bound ◽  
Nonempty Subset ◽  
Dominating Set ◽  
Domination Number ◽  
Extremal Graphs ◽  
Minimum Cardinality ◽  
Point Set

A set [Formula: see text] is a [Formula: see text]-point set dominating set (2-psd set) of a graph [Formula: see text] if for any subset [Formula: see text], there exists a nonempty subset [Formula: see text] containing at most two vertices such that the subgraph [Formula: see text] induced by [Formula: see text] is connected. The [Formula: see text]-point set domination number of [Formula: see text], denoted by [Formula: see text], is the minimum cardinality of a 2-psd set of [Formula: see text]. In this paper, we determine the lower bounds and an upper bound on [Formula: see text] of a graph. We also characterize extremal graphs for the lower bounds and identify some well-known classes of both separable and nonseparable graphs attaining the upper bound.

scholarly journals Some Results on Point Set Domination of Fuzzy Graphs

2013 ◽  
Vol 13 (2) ◽  
pp. 58-62
Author(s):  
S. Vimala ◽  
J. S. Sathya
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Fuzzy Graph ◽  
Minimal Point ◽  
Minimum Cardinality ◽  
Fuzzy Graphs ◽  
Point Set

Abstract Let G be a fuzzy graph. Let γ(G), γp(G) denote respectively the domination number, the point set domination number of a fuzzy graph. A dominating set D of a fuzzy graph is said to be a point set dominating set of a fuzzy graph if for every S⊆V-D there exists a node d∈D such that 〈S ∪ {d}〉 is a connected fuzzy graph. The minimum cardinality taken over all minimal point set dominating set is called a point set domination number of a fuzzy graph G and it is denoted by γp(G). In this paper we concentrate on the point set domination number of a fuzzy graph and obtain some bounds using the neighbourhood degree of fuzzy graphs.

scholarly journals Total Domination and Matching Numbers in Claw-Free Graphs

10.37236/1085 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Michael A. Henning ◽  
Anders Yeo
Keyword(s):  
Minimum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Matching Number ◽  
Total Domination ◽  
Total Domination Number ◽  
Free Graph ◽  
Induced Subgraph ◽  
Minimum Cardinality ◽  
Total Dominating Set

A set $M$ of edges of a graph $G$ is a matching if no two edges in $M$ are incident to the same vertex. The matching number of $G$ is the maximum cardinality of a matching of $G$. A set $S$ of vertices in $G$ is a total dominating set of $G$ if every vertex of $G$ is adjacent to some vertex in $S$. The minimum cardinality of a total dominating set of $G$ is the total domination number of $G$. If $G$ does not contain $K_{1,3}$ as an induced subgraph, then $G$ is said to be claw-free. We observe that the total domination number of every claw-free graph with minimum degree at least three is bounded above by its matching number. In this paper, we use transversals in hypergraphs to characterize connected claw-free graphs with minimum degree at least three that have equal total domination and matching numbers.

scholarly journals Bounds on Co-Independent Liar’s Domination in Graphs

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
K. Suriya Prabha ◽  
S. Amutha ◽  
N. Anbazhagan ◽  
Ismail Naci Cangul
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Induced Subgraph ◽  
Minimum Cardinality ◽  
Middle Graph ◽  
Domination In Graphs

A set S ⊆ V of a graph G = V , E is called a co-independent liar’s dominating set of G if (i) for all v ∈ V , N G v ∩ S ≥ 2 , (ii) for every pair u , v ∈ V of distinct vertices, N G u ∪ N G v ∩ S ≥ 3 , and (iii) the induced subgraph of G on V − S has no edge. The minimum cardinality of vertices in such a set is called the co-independent liar’s domination number of G , and it is denoted by γ coi L R G . In this paper, we introduce the concept of co-independent liar’s domination number of the middle graph of some standard graphs such as path and cycle graphs, and we propose some bounds on this new parameter.

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.

scholarly journals Complete Cototal Domination Number of a Graph

2011 ◽  
Vol 3 (3) ◽  
pp. 547-555 ◽  
Author(s):  
B. Basavanagoud ◽  
S. M. Hosamani
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Total Domination ◽  
Total Domination Number ◽  
Open Problems ◽  
Induced Subgraph ◽  
Minimum Cardinality ◽  
Total Dominating Set ◽  
Domination In Graphs

Let  be a graph. A set  of a graph  is called a total dominating set if the induced subgraph  has no isolated vertices. The total domination number  of G is the minimum cardinality of a total dominating set of G. A total dominating set D is said to be a complete cototal dominating set if the induced subgraph  has no isolated vertices. The complete cototal domination number  of G is the minimum cardinality of a complete cototal dominating set of G. In this paper, we initiate the study of complete cototal domination in graphs and present bounds and some exact values for . Also its relationship with other domination parameters are established and related two open problems are explored.Keywords: Domination number; Total domination number; Cototal domination number; Complete cototal domination number.© 2011 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved.doi:10.3329/jsr.v3i3.7744               J. Sci. Res. 3 (3), 557-565 (2011)

scholarly journals Block Split Subdivision Domination in Graphs

2015 ◽  
Vol 7 (3) ◽  
pp. 43-51
Author(s):  
M. H. Muddebihal ◽  
P Shekanna ◽  
S. Ahmed
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Induced Subgraph ◽  
Minimum Cardinality ◽  
Domination In Graphs

A dominating set D ? V[BS(G)] is a split dominating set in [BS(G)] if the induced subgraph  ?V[BS(G)] - D? is disconnected in [BS(G)]. The split domination number of [BS(G)] is denoted by ?sbs(G), is the minimum cardinality of a split dominating  set in   [BS(G)]. In this paper, some results on ?sbs(G) were obtained interms of vertices, blocks and other different parameters of G but not the members of [BS(G)]. Further we develop its relationship with other different domination parameters of G. 

scholarly journals Bounding the paired-domination number of a tree in terms of its annihilation number

Filomat ◽  
2014 ◽  
Vol 28 (3) ◽  
pp. 523-529 ◽  
Author(s):  
Nasrin Dehgardi ◽  
Seyed Sheikholeslami ◽  
Abdollah Khodkar
Keyword(s):  
Perfect Matching ◽  
Dominating Set ◽  
Degree Sequence ◽  
Domination Number ◽  
Induced Subgraph ◽  
Minimum Cardinality ◽  
Paired Domination Number ◽  
Isolated Vertex ◽  
Paired Domination

A paired-dominating set of a graph G=(V, E) with no isolated vertex is a dominating set of vertices whose induced subgraph has a perfect matching. The paired-domination number of G, denoted by ?pr(G), is the minimum cardinality of a paired-dominating set of G. The annihilation number a(G) is the largest integer k such that the sum of the first k terms of the non-decreasing degree sequence of G is at most the number of edges in G. In this paper, we prove that for any tree T of order n?2,?pr(T)? 4a(T)+2/3 and we characterize the trees achieving this bound.

Characterization of some classes of graphs with equal domination number and isolate domination number

2020 ◽  
Vol 12 (05) ◽  
pp. 2050065
Author(s):  
Davood Bakhshesh
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Interval Graph ◽  
Induced Subgraph ◽  
Minimum Cardinality ◽  
Vertex Set ◽  
Isolated Vertex ◽  
Proper Interval Graph ◽  
Domination In Graphs

Let [Formula: see text] be a simple and undirected graph with vertex set [Formula: see text]. A set [Formula: see text] is called a dominating set of [Formula: see text], if every vertex in [Formula: see text] is adjacent to at least one vertex in [Formula: see text]. The minimum cardinality of a dominating set of [Formula: see text] is called the domination number of [Formula: see text], denoted by [Formula: see text]. A dominating set [Formula: see text] of [Formula: see text] is called isolate dominating, if the induced subgraph [Formula: see text] of [Formula: see text] contains at least one isolated vertex. The minimum cardinality of an isolate dominating set of [Formula: see text] is called the isolate domination number of [Formula: see text], denoted by [Formula: see text]. In this paper, we show that for every proper interval graph [Formula: see text], [Formula: see text]. Moreover, we provide a constructive characterization for trees with equal domination number and isolate domination number. These solve part of an open problem posed by Hamid and Balamurugan [Isolate domination in graphs, Arab J. Math. Sci. 22(2) (2016) 232–241].

A lower bound on the total vertex-edge domination number of a tree

2020 ◽  
pp. 2050091
Author(s):  
B. Senthilkumar ◽  
H. Naresh Kumar ◽  
Y. B. Venkatakrishnan
Keyword(s):  
Lower Bound ◽  
Dominating Set ◽  
Domination Number ◽  
Edge Incident ◽  
Induced Subgraph ◽  
Minimum Cardinality ◽  
Edge Dominating Set

A vertex [Formula: see text] of a graph [Formula: see text] is said to vertex-edge dominate every edge incident to [Formula: see text], as well as every edge adjacent to these incident edges. A subset [Formula: see text] is a vertex-edge dominating set (ve-dominating set) if every edge of [Formula: see text] is vertex-edge dominated by at least one vertex of [Formula: see text]. A vertex-edge dominating set is said to be total if its induced subgraph has no isolated vertices. The minimum cardinality of a total vertex-edge dominating set of [Formula: see text], denoted by [Formula: see text], is called the total vertex-edge domination number of [Formula: see text]. In this paper, we prove that for every nontrivial tree of order [Formula: see text], with [Formula: see text] leaves and [Formula: see text] support vertices we have [Formula: see text], and we characterize extremal trees attaining the lower bound.

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