Restrained geodetic domination in graphs

2020 ◽  
Vol 12 (06) ◽  
pp. 2050084
Author(s):  
John Joy Mulloor ◽  
V. Sangeetha
Keyword(s):  
Connected Graph ◽  
Dominating Set ◽  
Cartesian Product ◽  
Minimum Cardinality ◽  
Geodetic Set ◽  
Vertex Set ◽  
Isolated Vertex ◽  
Join Of Graphs ◽  
Domination In Graphs ◽  
Corona Product

Let [Formula: see text] be a graph with edge set [Formula: see text] and vertex set [Formula: see text]. For a connected graph [Formula: see text], a vertex set [Formula: see text] of [Formula: see text] is said to be a geodetic set if every vertex in [Formula: see text] lies in a shortest path between any pair of vertices in [Formula: see text]. If the geodetic set [Formula: see text] is dominating, then [Formula: see text] is geodetic dominating set. A vertex set [Formula: see text] of [Formula: see text] is said to be a restrained geodetic dominating set if [Formula: see text] is geodetic, dominating and the subgraph induced by [Formula: see text] has no isolated vertex. The minimum cardinality of such set is called restrained geodetic domination (rgd) number. In this paper, rgd number of certain classes of graphs and 2-self-centered graphs was discussed. The restrained geodetic domination is discussed in graph operations such as Cartesian product and join of graphs. Restrained geodetic domination in corona product between a general connected graph and some classes of graphs is also discussed in this paper.

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.

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].

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.

General reduced second Zagreb index of graph operations

2020 ◽  
pp. 2150082
Author(s):  
R. Khoeilar ◽  
A. Jahanbani
Keyword(s):  
Real Number ◽  
Cartesian Product ◽  
Zagreb Index ◽  
Second Zagreb Index ◽  
Graph Operations ◽  
Vertex Set ◽  
Join Of Graphs ◽  
Corona Product

Let [Formula: see text] be a graph with vertex set [Formula: see text] and edge set [Formula: see text]. The general reduced second Zagreb index of [Formula: see text] is defined as [Formula: see text], where [Formula: see text] is any real number and [Formula: see text] is the degree of the vertex [Formula: see text] of [Formula: see text]. In this paper, the general reduced second Zagreb index of the Cartesian product, corona product, join of graphs and two new operations of graphs are computed.

scholarly journals The edge-to-edge geodetic domination number of a graph

2021 ◽  
Vol 40 (3) ◽  
pp. 635-658
Author(s):  
J. John ◽  
V. Sujin Flower
Keyword(s):  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Upper Bounds ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
General Properties ◽  
Geodetic Number ◽  
Geodetic Set ◽  
Positive Integers

Let G = (V, E) be a connected graph with at least three vertices. A set S ⊆ E(G) is called an edge-to-edge geodetic dominating set of G if S is both an edge-to-edge geodetic set of G and an edge dominating set of G. The edge-to-edge geodetic domination number γgee(G) of G is the minimum cardinality of its edge-to-edge geodetic dominating sets. Some general properties satisfied by this concept are studied. Connected graphs of size m with edge-to-edge geodetic domination number 2 or m or m − 1 are characterized. We proved that if G is a connected graph of size m ≥ 4 and Ḡ is also connected, then 4 ≤ γgee(G) + γgee(Ḡ) ≤ 2m − 2. Moreover we characterized graphs for which the lower and the upper bounds are sharp. It is shown that, for every pair of positive integers a, b with 2 ≤ a ≤ b, there exists a connected graph G with gee(G) = a and γgee(G) = b. Also it is shown that, for every pair of positive integers a and b with 2 < a ≤ b, there exists a connected graph G with γe(G) = a and γgee(G) = b, where γe(G) is the edge domination number of G and gee(G) is the edge-to-edge geodetic number of G.

scholarly journals A Comparison on the Bounds of Chromatic Preserving Number and Dom-Chromatic Number of Cartesian Product and Kronecker Product of Paths

2012 ◽  
Vol 11 (4) ◽  
pp. 43-58
Author(s):  
T N Janakiraman ◽  
M Poobalaranjani
Keyword(s):  
Comparative Study ◽  
Chromatic Number ◽  
Kronecker Product ◽  
Dominating Set ◽  
Cartesian Product ◽  
Simple Graph ◽  
Minimum Cardinality ◽  
Vertex Set

Let G be a simple graph with vertex set V and edge set E. A Set S Í V is said to be a chromatic preserving set or a cp-set if χ(<S>) = χ(G) and the minimum cardinality of a cp-set in G is called the chromatic preserving number or cp-number of G and is denoted by cpn(G). A cp-set of cardinality cpn(G) is called a cpn-set. A subset S of V is said to be a dom- chromatic set (or a dc-set) if S is a dominating set and χ(<S>) = χ(G). The minimum cardinality of a dom-chromatic set in a graph G is called the dom-chromatic number (or dc- number) of G and is denoted by γch(G). The Kronecker product G1 Ù G2 of two graphs G1 = (V1, E1) and G2 = (V2, E2) is the graph G with vertex set V1 x V2 and any two distinct vertices (u1, v1) and (u2, v2) of G are adjacent if u1u2 Î E1 and v1v2 Î E2. The Cartesian product G1 x G2 is the graph with vertex set V1 x V2 where any two distinct vertices (u1, v1) and (u2, v2) are adjacent whenever (i) u1 = u2 and v1v2 Î E2 or (ii) u1u2 Î E1 and v1 = v2. These two products have no common edges. They are almost like complements but not exactly. In this paper, we discuss the behavior of the cp-number and dc-number and their bounds for product of paths in the two cases. A detailed comparative study is also done.

Some results for the two disjoint connected dominating sets problem

2019 ◽  
Vol 11 (06) ◽  
pp. 1950065
Author(s):  
Xianliang Liu ◽  
Zishen Yang ◽  
Wei Wang
Keyword(s):  
Upper Bound ◽  
Connected Graph ◽  
Dominating Set ◽  
Sufficient Condition ◽  
Dominating Sets ◽  
Cubic Graph ◽  
Connected Dominating Sets ◽  
Minimum Cardinality ◽  
Vertex Set ◽  
Dominating Set Problem

As a variant of minimum connected dominating set problem, two disjoint connected dominating sets (DCDS) problem is to ask whether there are two DCDS [Formula: see text] in a connected graph [Formula: see text] with [Formula: see text] and [Formula: see text], and if not, how to add an edge subset with minimum cardinality such that the new graph has a pair of DCDS. The two DCDS problem is so hard that it is NP-hard on trees. In this paper, if the vertex set [Formula: see text] of a connected graph [Formula: see text] can be partitioned into two DCDS of [Formula: see text], then it is called a DCDS graph. First, a necessary but not sufficient condition is proposed for cubic (3-regular) graph to be a DCDS graph. To be exact, if a cubic graph is a DCDS graph, there are at most four disjoint triangles in it. Next, if a connected graph [Formula: see text] is a DCDS graph, a simple but nontrivial upper bound [Formula: see text] of the girth [Formula: see text] is presented.

Vertex criticality with respect to isolate domination

2015 ◽  
Vol 07 (02) ◽  
pp. 1550010 ◽  
Author(s):  
I. Sahul Hamid ◽  
S. Balamurugan
Keyword(s):  
Maximum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Minimum Cardinality ◽  
Unicyclic Graphs ◽  
Extended Chain ◽  
Isolated Vertex ◽  
Domination In Graphs

A set S of vertices of a graph G is called a dominating set of G if every vertex in V(G) - S is adjacent to a vertex in S. A dominating set S such that the subgraph 〈S〉 induced by S has at least one isolated vertex is called an isolate dominating set. The minimum cardinality of an isolate dominating set is called the isolate domination number and is denoted by γ0(G). This concept was introduced in [I. Sahul Hamid and S. Balamurugan, Isolate domination in graphs (Communicated)] and further studied in [I. Sahul Hamid and S. Balamurugan, Extended chain of domination parameters in graphs, ISRN Combin. 2013 (2013), Article ID: 792743, 4 pp.; Isolate domination and maximum degree, Bull. Int. Math. Virtual Inst. 3 (2013) 127–133; Isolate domination in unicyclic graphs, Int. J. Math. Soft Comput. 3(3) (2013) 79–83]. This paper studies the effect of the removal of a vertex upon the isolate domination number.

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