scholarly journals Restrained Edge Domination Number of Some Path Related Graphs

2021 ◽  
Vol 13 (1) ◽  
pp. 145-151
Author(s):  
S. K. Vaidya ◽  
P. D. Ajani
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Minimum Cardinality ◽  
Restrained Domination ◽  
Edge Dominating Set ◽  
Middle Graph

For a graph G = (V,E), a set  S ⊆ V(S ⊆ E) is a restrained dominating (restrained edge dominating) set if every vertex (edge) not in S is adjacent (incident) to a vertex (edge) in S and to a vertex (edge) in V - S(E-S). The minimum cardinality of a restrained dominating (restrained edge dominating) set of G is called restrained domination (restrained edge domination) number of G, denoted by  γr (G) (γre(G). The restrained edge domination number of some standard graphs are already investigated while in this paper the restrained edge domination number like degree splitting, switching,  square and middle graph obtained from path.

scholarly journals On Two Open Problems on Double Vertex-Edge Domination in Graphs

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1010
Author(s):  
Fang Miao ◽  
Wenjie Fan ◽  
Mustapha Chellali ◽  
Rana Khoeilar ◽  
Seyed Mahmoud Sheikholeslami ◽  
...  
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Total Domination ◽  
Total Domination Number ◽  
Open Problems ◽  
Free Graph ◽  
Minimum Cardinality ◽  
Total Dominating Set ◽  
Edge Dominating Set ◽  
Domination In Graphs

A vertex v of a graph G = ( V , E ) , ve-dominates every edge incident to v, as well as every edge adjacent to these incident edges. A set S ⊆ V is a double vertex-edge dominating set if every edge of E is ve-dominated by at least two vertices of S. The double vertex-edge domination number γ d v e ( G ) is the minimum cardinality of a double vertex-edge dominating set in G. A subset S ⊆ V is a total dominating set (respectively, a 2-dominating set) if every vertex in V has a neighbor in S (respectively, every vertex in V - S has at least two neighbors in S). The total domination number γ t ( G ) is the minimum cardinality of a total dominating set of G, and the 2-domination number γ 2 ( G ) is the minimum cardinality of a 2-dominating set of G . Krishnakumari et al. (2017) showed that for every triangle-free graph G , γ d v e ( G ) ≤ γ 2 ( G ) , and in addition, if G has no isolated vertices, then γ d v e ( G ) ≤ γ t ( G ) . Moreover, they posed the problem of characterizing those graphs attaining the equality in the previous bounds. In this paper, we characterize all trees T with γ d v e ( T ) = γ t ( T ) or γ d v e ( T ) = γ 2 ( T ) .

Critical concepts of restrained domination in signed graphs

2021 ◽  
pp. 2250010
Author(s):  
Anisha Jean Mathias ◽  
V. Sangeetha ◽  
Mukti Acharya
Keyword(s):  
Undirected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Signed Graph ◽  
Signed Graphs ◽  
Minimum Cardinality ◽  
Underlying Graph ◽  
Restrained Domination ◽  
Edge Removal ◽  
Restrained Domination Number

A signed graph [Formula: see text] is a simple undirected graph in which each edge is either positive or negative. Restrained dominating set [Formula: see text] in [Formula: see text] is a restrained dominating set of the underlying graph [Formula: see text] where the subgraph induced by the edges across [Formula: see text] and within [Formula: see text] is balanced. The minimum cardinality of a restrained dominating set of [Formula: see text] is called the restrained domination number, denoted by [Formula: see text]. In this paper, we initiate the study on various critical concepts to investigate the effect of edge removal or edge addition on restrained domination number in signed graphs.

Bounds on total edge domination number of a tree

2020 ◽  
pp. 2150011
Author(s):  
B. Senthilkumar ◽  
H. Naresh Kumar ◽  
Y. B. Venkatakrishnan
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Minimum Cardinality ◽  
Edge Dominating Set ◽  
Vertex Set

For a 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 the total edge dominating set if every edge in [Formula: see text] is adjacent to at least one edge in [Formula: see text]. The minimum cardinality of a total edge dominated set, denoted by [Formula: see text], is called the total edge domination number of a graph [Formula: see text]. We prove that for every tree [Formula: see text] of diameter at least two with [Formula: see text] leaves and [Formula: see text] support vertices we have [Formula: see text], and we characterize the trees attaining each of the bounds.

scholarly journals Edge Domination in Some Path and Cycle Related Graphs

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
S. K. Vaidya ◽  
R. M. Pandit
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
Edge Dominating Set

For a graph G=V,E, a subset F of E is called an edge dominating set of G if every edge not in F is adjacent to some edge in F. The edge domination number γ′G of G is the minimum cardinality taken over all edge dominating sets of G. Here, we determine the edge domination number for shadow graphs, middle graphs, and total graphs of paths and cycles.

scholarly journals DOMINATION AND EDGE DOMINATION IN TREES

2020 ◽  
Vol 6 (1) ◽  
pp. 147
Author(s):  
B. Senthilkumar ◽  
Yanamandram B. Venkatakrishnan ◽  
H. Naresh Kumar
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Simple Graph ◽  
Minimum Cardinality ◽  
Edge Dominating Set

Let \(G=(V,E)\) be a simple graph. A set \(S\subseteq V\) is a dominating set if every vertex in \(V \setminus S\) is adjacent to a vertex in \(S\). The domination number of a graph \(G\), denoted by \(\gamma(G)\) is the minimum cardinality of a dominating set of \(G\). A set \(D \subseteq E\) is an edge dominating set if every edge in \(E\setminus D\) is adjacent to an edge in \(D\). The edge domination number of a graph \(G\), denoted by \(\gamma'(G)\) is the minimum cardinality of an edge dominating set of \(G\). We characterize trees with  domination number equal to twice edge domination number.

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.

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.

Double vertex-edge domination

2017 ◽  
Vol 09 (04) ◽  
pp. 1750045 ◽  
Author(s):  
Balakrishna Krishnakumari ◽  
Mustapha Chellali ◽  
Yanamandram B. Venkatakrishnan
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Dominating Set ◽  
Domination Number ◽  
Edge Incident ◽  
Minimum Cardinality ◽  
Edge Dominating Set ◽  
Number Formula ◽  
Number Of Leaves ◽  
Np Complete

A vertex [Formula: see text] of a graph [Formula: see text] is said to [Formula: see text]-dominate every edge incident to [Formula: see text], as well as every edge adjacent to these incident edges. A set [Formula: see text] is a vertex-edge dominating set (double vertex-edge dominating set, respectively) if every edge of [Formula: see text] is [Formula: see text]-dominated by at least one vertex (at least two vertices) of [Formula: see text] The minimum cardinality of a vertex-edge dominating set (double vertex-edge dominating set, respectively) of [Formula: see text] is the vertex-edge domination number [Formula: see text] (the double vertex-edge domination number [Formula: see text], respectively). In this paper, we initiate the study of double vertex-edge domination. We first show that determining the number [Formula: see text] for bipartite graphs is NP-complete. We also prove that for every nontrivial connected graphs [Formula: see text] [Formula: see text] and we characterize the trees [Formula: see text] with [Formula: see text] or [Formula: see text] Finally, we provide two lower bounds on the double ve-domination number of trees and unicycle graphs in terms of the order [Formula: see text] the number of leaves and support vertices, and we characterize the trees attaining the lower bound.

scholarly journals Neighborhood connected edge domination in graphs

2012 ◽  
Vol 43 (1) ◽  
pp. 69-80
Author(s):  
Kulandaivel M.P. ◽  
C. Sivagnanam ◽  
P. Selvaraju
Keyword(s):  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Induced Subgraph ◽  
Minimum Cardinality ◽  
Edge Dominating Set ◽  
Domination In Graphs

Let G = (V,E) be a connected graph. An edge dominating set X of G is called a neighborhood connected edge dominating set (nced-set) if the edge induced subgraph < N(X) > is connected. The minimum cardinality of a nced-set of G is called the neighborhood connected edge domination number of G and is denoted by. In this paper we initiate a study of this parameter.

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