Split Domination Number in Edge Semi-Middle Graph

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.

Transactions on Combinatorics, 2014

Given a graph $G = (V,E)$, a dominating set $D subseteq� V$ is called a semi-strong split dominating set of $G$ if $|V setminus D| geq� 1$ and the maximum degree of the subgraph induced by $V setminus D$ is 1. The minimum cardinality of a semi-strong split dominating set (SSSDS) of G is the semi-strong split domination number of G, denoted $gamma_{sss}(G)$. In this work, we introduce the concept and prove several results regarding it.

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.

IRJMETS Publication, 2021

A set of vertices in a splitted graph () = (,) is called a dominating set if every vertex in − is adjacent to some vertex in. A dominating set is said to be a connected dominating set if the subgraph < > induced is connected in (). The connected domination number [ ()] of () is minimum cardinality of a Connected domination set of ().

International journal of statistics and applied mathematics, 2022

Let (,) be a connected graph and () be its corresponding vertex semi middle graph. A dominating set ⊆ [ ()] is Nonsplit dominating set 〈 [ ()] − 〉 is connected. The minimum size of D is called the Nonsplit domination number of () and is denoted by [ ()]. In this paper we obtain several results on Nonsplit domination number.

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.

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.

In a graph G, a vertex dominates itself and its neighbours. A subset S of V is called a dominating set in G if every vertex in V is dominated by at least one vertex in S. The domination number  (G) is the minimum cardinality of a dominating set. A set S V is called a double dominating set of a graph G if every vertex in V is dominated by at least two vertices in S. The minimum cardinality of a double dominating set is called double domination number of G and is denoted by dd(G). The connectivity  (G) of a connected graph G is the minimum number of vertices whose removal results in a disconnected or trivial graph. In this paper we find an upper bound for the sum of the double domination number and connectivity of a graph and characterize the corresponding extremal graphs.

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.