scholarly journals Power domination in splitting and degree splitting graph

2021 ◽  
Vol 40 (6) ◽  
pp. 1641-1655
Author(s):  
J. Anitha ◽  
S. Muthukumar
Keyword(s):  
Dominating Set ◽  
Power Grid ◽  
Domination Number ◽  
Power Domination ◽  
Grid Monitoring ◽  
Vertex Set ◽  
Splitting Graph

A vertex set S is called a power dominating set of a graph G if every vertex within the system is monitored by the set S following a collection of rules for power grid monitoring. The power domination number of G is the order of a minimal power dominating set of G. In this paper, we solve the power domination number for splitting and degree splitting graph.

scholarly journals Equitable Power Domination Number of Mycielskian of Certain Graphs

2018 ◽  
Vol 7 (4.10) ◽  
pp. 842 ◽  
Author(s):  
S. Banu Priya ◽  
A. Parthiban ◽  
N. Srinivasan
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Simple Graph ◽  
Adjacent Vertex ◽  
Minimum Cardinality ◽  
Power Domination ◽  
Vertex Set ◽  
The Difference

Let  be a simple graph with vertex set  and edge set . A set  is called a power dominating set (PDS), if every vertex   is observed by some vertices in  by using the following rules: (i) if a vertex  in  is in PDS, then it dominates itself and all the adjacent vertices of  and (ii) if an observed vertex  in   has  adjacent vertices and if   of these vertices are already observed, then the remaining one non-observed vertex is also observed by  in . A power dominating set    in   is said to be an equitable power dominating set (EPDS), if for every  there exists an adjacent vertex   such that the difference between the degree of  and degree of  is less than or equal to 1, i.e., . The minimum cardinality of an equitable power dominating set of  is called the equitable power domination number of  and denoted by . The Mycielskian of a graph  is the graph  with vertex set  where , and edge set  In this paper we investigate the equitable power domination number of Mycielskian of certain graphs. 

scholarly journals Equitable Power Domination Number of Certain Graphs

2018 ◽  
Vol 7 (4.10) ◽  
pp. 349 ◽  
Author(s):  
S. Banu Priya ◽  
N. Srinivasan
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Power Domination ◽  
Vertex Set

Let   12G">  be a graph with vertex set 12V"> , a set 12Sâٹ†V">  is said to be a power dominating set (PDS), if every vertex  12u∈V-S">  is observed by some vertices in 12S">  using the following rules: (i) if a vertex  12v">  in 12G">  is in PDS, then it dominates itself and all the adjacent vertices of 12v">  and (ii) if an observed vertex 12v">  in 12G">  has  12k>1">  adjacent vertices and if 12k-1">  of these vertices are already observed, then the remaining one non-observed vertex will also be observed by 12v">  in 12G"> . The degree 12d(v)">  of a vertex 12v">  in 12G">  is the number of edges of 12G">  incident with 12v">  and any two adjacent vertices 12u">  and 12v">  in 12G">  are said to hold equitable property if 12|d(u)-d(v)| ≤ 1"> . In this paper, we introduce the notions of equitable power dominating set and equitable power domination number. We also derive the equitable power domination number of certain graphs.  

scholarly journals Bipartite graphs with close domination and k-domination numbers

2020 ◽  
Vol 18 (1) ◽  
pp. 873-885
Author(s):  
Gülnaz Boruzanlı Ekinci ◽  
Csilla Bujtás
Keyword(s):  
Positive Integer ◽  
Dominating Set ◽  
Domination Number ◽  
Bipartite Graphs ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Minimum Cardinality ◽  
Vertex Set ◽  
Necessary And Sufficient ◽  
The Given

Abstract Let k be a positive integer and let G be a graph with vertex set V(G) . A subset D\subseteq V(G) is a k -dominating set if every vertex outside D is adjacent to at least k vertices in D . The k -domination number {\gamma }_{k}(G) is the minimum cardinality of a k -dominating set in G . For any graph G , we know that {\gamma }_{k}(G)\ge \gamma (G)+k-2 where \text{Δ}(G)\ge k\ge 2 and this bound is sharp for every k\ge 2 . In this paper, we characterize bipartite graphs satisfying the equality for k\ge 3 and present a necessary and sufficient condition for a bipartite graph to satisfy the equality hereditarily when k=3 . We also prove that the problem of deciding whether a graph satisfies the given equality is NP-hard in general.

Complementary equitably totally disconnected equitable domination in graphs

2020 ◽  
pp. 2150043
Author(s):  
P. Nataraj ◽  
R. Sundareswaran ◽  
V. Swaminathan
Keyword(s):  
Undirected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Minimum Cardinality ◽  
Vertex Set ◽  
Domination In Graphs ◽  
Totally Disconnected

In a simple, finite and undirected graph [Formula: see text] with vertex set [Formula: see text] and edge set [Formula: see text], a subset [Formula: see text] of [Formula: see text] is said to be a degree equitable dominating set if for every [Formula: see text] there exists a vertex [Formula: see text] such that [Formula: see text] and [Formula: see text], where [Formula: see text] denotes the degree of [Formula: see text] in [Formula: see text]. The minimum cardinality of such a dominating set is denoted by [Formula: see text] and is called the equitable domination number of [Formula: see text]. In this paper, we introduce Complementary Equitably Totally Disconnected Equitable domination in graphs and obtain some interesting results. Also, we discuss some bounds of this new domination parameter.

Outer-paired domination in graphs

2020 ◽  
Vol 12 (06) ◽  
pp. 2050072
Author(s):  
A. Mahmoodi ◽  
L. Asgharsharghi
Keyword(s):  
Perfect Matching ◽  
Dominating Set ◽  
Domination Number ◽  
Simple Graph ◽  
Minimum Cardinality ◽  
Sharp Bounds ◽  
Paired Domination Number ◽  
Vertex Set ◽  
Paired Domination ◽  
Domination In Graphs

Let [Formula: see text] be a simple graph with vertex set [Formula: see text] and edge set [Formula: see text]. An outer-paired dominating set [Formula: see text] of a graph [Formula: see text] is a dominating set such that the subgraph induced by [Formula: see text] has a perfect matching. The outer-paired domination number of [Formula: see text], denoted by [Formula: see text], is the minimum cardinality of an outer-paired dominating set of [Formula: see text]. In this paper, we study the outer-paired domination number of graphs and present some sharp bounds concerning the invariant. Also, we characterize all the trees with [Formula: see text].

Disjunctive total domination in permutation graphs

2017 ◽  
Vol 09 (01) ◽  
pp. 1750009 ◽  
Author(s):  
Eunjeong Yi
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Total Domination ◽  
Permutation Graph ◽  
Permutation Graphs ◽  
Minimum Cardinality ◽  
Total Dominating Set ◽  
Vertex Set ◽  
Isolated Vertex ◽  
Number Formula

Let [Formula: see text] be a graph with vertex set [Formula: see text] and edge set [Formula: see text]. If [Formula: see text] has no isolated vertex, then a disjunctive total dominating set (DTD-set) of [Formula: see text] is a vertex set [Formula: see text] such that every vertex in [Formula: see text] is adjacent to a vertex of [Formula: see text] or has at least two vertices in [Formula: see text] at distance two from it, and the disjunctive total domination number [Formula: see text] of [Formula: see text] is the minimum cardinality overall DTD-sets of [Formula: see text]. Let [Formula: see text] and [Formula: see text] be two disjoint copies of a graph [Formula: see text], and let [Formula: see text] be a bijection. Then, a permutation graph [Formula: see text] has the vertex set [Formula: see text] and the edge set [Formula: see text]. For any connected graph [Formula: see text] of order at least three, we prove the sharp bounds [Formula: see text]; we give an example showing that [Formula: see text] can be arbitrarily large. We characterize permutation graphs for which [Formula: see text] holds. Further, we show that [Formula: see text] when [Formula: see text] is a cycle, a path, and a complete [Formula: see text]-partite graph, respectively.

scholarly journals Graphs with Constant Sum of Domination and Inverse Domination Numbers

2012 ◽  
Vol 2012 ◽  
pp. 1-7 ◽  
Author(s):  
T. Tamizh Chelvam ◽  
T. Asir
Keyword(s):  
Minimum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Complete Characterization ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
Connected Graphs ◽  
Vertex Set ◽  
Domination Numbers

A subset D of the vertex set of a graph G, is a dominating set if every vertex in V−D is adjacent to at least one vertex in D. The domination number γ(G) is the minimum cardinality of a dominating set of G. A subset of V−D, which is also a dominating set of G is called an inverse dominating set of G with respect to D. The inverse domination number γ′(G) is the minimum cardinality of the inverse dominating sets. Domke et al. (2004) characterized connected graphs G with γ(G)+γ′(G)=n, where n is the number of vertices in G. It is the purpose of this paper to give a complete characterization of graphs G with minimum degree at least two and γ(G)+γ′(G)=n−1.

scholarly journals Edge stability in secure graph domination

2015 ◽  
Vol Vol. 17 no. 1 (Graph Theory) ◽  
Author(s):  
Anton Pierre Burger ◽  
Alewyn Petrus Villiers ◽  
Jan Harm Vuuren
Keyword(s):  
Graph Theory ◽  
Dominating Set ◽  
Domination Number ◽  
Arbitrary Subset ◽  
Graph Domination ◽  
Vertex Set ◽  
International Audience ◽  
Edge Stability ◽  
Secure Domination

Graph Theory International audience A subset X of the vertex set of a graph G is a secure dominating set of G if X is a dominating set of G and if, for each vertex u not in X, there is a neighbouring vertex v of u in X such that the swap set (X-v)∪u is again a dominating set of G. The secure domination number of G is the cardinality of a smallest secure dominating set of G. A graph G is p-stable if the largest arbitrary subset of edges whose removal from G does not increase the secure domination number of the resulting graph, has cardinality p. In this paper we study the problem of computing p-stable graphs for all admissible values of p and determine the exact values of p for which members of various infinite classes of graphs are p-stable. We also consider the problem of determining analytically the largest value ωn of p for which a graph of order n can be p-stable. We conjecture that ωn=n-2 and motivate this conjecture.

scholarly journals A note on bipartite graphs whose [1,k]-domination number equal to their number of vertices

2020 ◽  
Vol 40 (3) ◽  
pp. 375-382
Author(s):  
Narges Ghareghani ◽  
Iztok Peterin ◽  
Pouyeh Sharifani
Keyword(s):  
Decision Problem ◽  
Dominating Set ◽  
Domination Number ◽  
Short Note ◽  
Bipartite Graphs ◽  
Hard Problem ◽  
Simple Construction ◽  
Minimum Number ◽  
Vertex Set ◽  
Np Hard Problem

A subset \(D\) of the vertex set \(V\) of a graph \(G\) is called an \([1,k]\)-dominating set if every vertex from \(V-D\) is adjacent to at least one vertex and at most \(k\) vertices of \(D\). A \([1,k]\)-dominating set with the minimum number of vertices is called a \(\gamma_{[1,k]}\)-set and the number of its vertices is the \([1,k]\)-domination number \(\gamma_{[1,k]}(G)\) of \(G\). In this short note we show that the decision problem whether \(\gamma_{[1,k]}(G)=n\) is an \(NP\)-hard problem, even for bipartite graphs. Also, a simple construction of a bipartite graph \(G\) of order \(n\) satisfying \(\gamma_{[1,k]}(G)=n\) is given for every integer \(n \geq (k+1)(2k+3)\).

scholarly journals Co-Secure Set Domination in Graphs

2020 ◽  
Vol 8 (4S5) ◽  
pp. 66-68
Keyword(s):  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Minimum Cardinality ◽  
Vertex Set ◽  
Secure Set ◽  
Domination In Graphs

Throughout this paper, consider G = (V,E) as a connected graph. A subset D of V(G) is a set dominating set of G if for every M  V / D there exists a non-empty set N of D such that the induced sub graph <MUN> is connected. A subset D of the vertex set of a graph G is called a co-secure dominating set of a graph if D is a dominating set, and for each u' D there exists a vertex v'V / D such that u'v' is an edge and D \u'v' is a dominating set. A co-secure dominating set D is a co-secure set dominating set of G if D is also a set dominating set of G. The co-secure set domination number G s cs γ is the minimum cardinality of a co-secure set dominating set. In this paper we initiate the study of this new parameter & also determine the co-secure set domination number of some standard graphs and obtain its bounds.

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