Semi-strong split domination in graphs

Journal of Discrete Mathematical Sciences and Cryptography, 2009

A dominating set D of a graph G is a split dominating set if the induced subgraph < V − D > is disconnected. The split domination number γ s (G) is the minimum cardinality of a split dominating set. The concept of split domination number was introduced by Kulli and Janakiram. In this paper, some results on split domination are obtained.

Let G=(V,E) be a simple, undirected, finite nontrivial graph. A non empty set SV of vertices in a graph G is called a dominating set if every vertex in V-S is adjacent to some vertex in S. The domination number γ(G) of G is the minimum cardinality of a dominating set of G.A dominating set S is called a non split set dominating set if there exists a non empty set R S such that <RT> is connected for every set TV-S and the induced subgraph <V-S> is not connected. The minimum cardinality of a split set dominating set is called the split set domination number of G and is denoted by γss (G). In this paper, bounds for γss (G) and exact values for some particular classes of graphs are found and also the split set domination number of some standard graphs is given in this paper.

2018

A nonempty set D V of a graph G is a dominating set of G if every vertex in V-D is adjacent to some vertex in D. The domination number  (G) is the minimum cardinality taken over all the minimal dominating sets of G. Let D be the minimum dominating set of G. If V-D contains a dominating set D then D is called the Inverse dominating set of G w.r.to D. The Inverse dominating number  (G) is the minimum cardinality taken over all the minimal inverse dominating sets of G. A dominating set D of G is a connected dominating set if the induced subgraph &lt;D&gt; is connected. The connected domination number c(G) is the minimum cardinality of a connected dominating set. Unless stated, the graph G has n vertices and m edges. A dominating set D V of a graph G is a split (non-split) dominating set if the induced subgraph &lt;V-D&gt; is disconnected (connected). The split (non-split) domination number s(G) (ns(G)) is the minimum cardinality of a split (non-split) dominating set.

2021

A set D of vertices in an isolate-free graph G is a semitotal dominating set of G if D is a dominating set of G and every vertex in D is within distance 2 from another vertex of D. The semitotal domination number of G is the minimum cardinality of a semitotal dominating set of G and is denoted by γt2(G). In this paper after computation of semitotal domination number of specific graphs, we count the number of this kind of dominating sets of arbitrary size in some graphs.

Pan-American Journal of Mathematics

Let G = (p, q) be a connected graph and Me(G) be its corresponding edge semi-middle graph. A dominating set D ⊆ V [Me(G)] is split dominating set V [Me(G)] – D is disconnected. The minimum size of D is called the split domination number of Me(G) and is denoted by γs[Me(G)]. In this paper we obtain several results on split domination number.

An equitable dominating set D of a graph G = (V, E) is a split equitable dominating set if the induced subgraph hV − Di is a disconnected. The split equitable domination number ϒse(G) of a graph G is the minimum cardinality of a split equitable dominating set. In this paper, we initiate the study of this new parameter and present some bounds and some exact values for ϒse(G). Also Nordhaus−Gaddum type results are obtained.

Indian Journal of Applied Research, 2011

2021

In this paper, we introduce and investigate some new splitted graphs called \(S(P_l); S(H_l); S(P^+_l ), S(P_loNK_1) .\)&lt;br&gt; Also we discuss some splitted graphs and it&#39;s properties are obtained.

International Journal of Mathematics Trends and Technology, 2014

A dominating set D of a splitted graph S(G) = ( V, E ) is an independent dominating set if the induced subgraph has no edges. The independent domination number i[S(G)] of a graph S(G) is the minimum cardinality of an independent dominating set.

Discussiones Mathematicae Graph Theory, 2016

A set S of vertices of a graph G is a dominating set if every vertex not in S is adjacent to a vertex of S and is a total dominating set if every vertex of G is adjacent to a vertex of S. The cardinality of a minimum dominating (total dominating) set of G is called the domination (total domination) number. A set that does not dominate (totally dominate) G is called a nondominating (non-total dominating) set of G. A partition of the vertices of G into non-dominating (non-total dominating) sets is a non-dominating (nontotal dominating) set partition. We show that the minimum number of sets in a non-dominating set partition of a graph G equals the total domination number of its complement G and the minimum number of sets in a non-total dominating set partition of G equals the domination number of G. This perspective yields new upper bounds on the domination and total domination numbers. We motivate the study of these concepts with a social network application.