scholarly journals On Locating-Dominating Set of Regular Graphs

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Anuwar Kadir Abdul Gafur ◽  
Suhadi Wido Saputro
Keyword(s):  
Regular Graph ◽  
Dominating Set ◽  
Finite Graph ◽  
Regular Graphs ◽  
Minimum Cardinality ◽  
Dominating Number

Let G be a simple, connected, and finite graph. For every vertex v ∈ V G , we denote by N G v the set of neighbours of v in G . The locating-dominating number of a graph G is defined as the minimum cardinality of W   ⊆   V G such that every two distinct vertices u , v ∈ V G \ W satisfies ∅ ≠ N G u ∩ W ≠ N G v ∩ W ≠ ∅ . A graph G is called k -regular graph if every vertex of G is adjacent to k other vertices of G . In this paper, we determine the locating-dominating number of k -regular graph of order n , where k = n − 2 or k = n − 3 .

scholarly journals On Triangular Secure Domination Number

2020 ◽  
Vol 2 (2) ◽  
pp. 105-110
Author(s):  
Emily L Casinillo ◽  
Leomarich F Casinillo ◽  
Jorge S Valenzona ◽  
Divina L Valenzona
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Triangular Grid ◽  
Combinatorial Formula ◽  
Grid Graph ◽  
Minimum Cardinality ◽  
Dominating Number ◽  
Triangular Number ◽  
Perfect Numbers ◽  
Secure Domination

Let T_m=(V(T_m), E(T_m)) be a triangular grid graph of m ϵ N level. The order of graph T_m is called a triangular number. A subset T of V(T_m) is a dominating set of T_m  if for all u_V(T_m)\T, there exists vϵT such that uv ϵ E(T_m), that is, N[T]=V(T_m).  A dominating set T of V(T_m) is a secure dominating set of T_m if for each u ϵ V(T_m)\T, there exists v ϵ T such that uv ϵ E(T_m) and the set (T\{u})ꓴ{v} is a dominating set of T_m. The minimum cardinality of a secure dominating set of T_m, denoted by γ_s(T_m)  is called a secure domination number of graph T_m. A secure dominating number  γ_s(T_m) of graph T_m is a triangular secure domination number if γ_s(T_m) is a triangular number. In this paper, a combinatorial formula for triangular secure domination number of graph T_m was constructed. Furthermore, the said number was evaluated in relation to perfect numbers.

Doubly connected bi-domination in graphs

2020 ◽  
pp. 2150009 ◽  
Author(s):  
Mohammed A. Abdlhusein
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Finite Graph ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
Isolated Vertex ◽  
Domination In Graphs

Let [Formula: see text] be a finite graph, simple, undirected and has no isolated vertex. A dominating subset [Formula: see text] of [Formula: see text] is said a bi-dominating set, if every vertex of it dominates two vertices of [Formula: see text]. The bi-domination number of [Formula: see text], denoted by [Formula: see text] is the minimum cardinality over all bi-dominating sets in [Formula: see text]. In this paper, a certain modified bi-domination parameter called doubly connected bi-domination and its inverse are introduced. Several bounds and properties are studied here. These modified dominations are applied and evaluated for several well-known graphs and complement graphs.

scholarly journals Computing the split domination number of grid graphs

2021 ◽  
Vol 5 (1) ◽  
pp. 1
Author(s):  
V. R. Girish ◽  
P. Usha
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
New Method ◽  
Minimum Cardinality ◽  
Grid Graphs ◽  
Star Graphs ◽  
Dominating Number ◽  
Vertex Set ◽  
Induced Graph

<p>A set <em>D</em> - <em>V</em> is a dominating set of <em>G</em> if every vertex in <em>V - D</em> is adjacent to some vertex in <em>D</em>. The dominating number γ(<em>G</em>) of <em>G</em> is the minimum cardinality of a dominating set <em>D</em>. A dominating set <em>D</em> of a graph <em>G</em> = (<em>V;E</em>) is a split dominating set if the induced graph (<em>V</em> - <em>D</em>) is disconnected. The split domination number γ<em><sub>s</sub></em>(<em>G</em>) is the minimum cardinality of a split domination set. In this paper we have introduced a new method to obtain the split domination number of grid graphs by partitioning the vertex set in terms of star graphs and also we have<br />obtained the exact values of γ<em>s</em>(<em>G<sub>m;n</sub></em>); <em>m</em> ≤ <em>n</em>; <em>m,n</em> ≤ 24:</p>

scholarly journals New Probabilistic Upper Bounds on the Domination Number of a Graph

10.37236/8345 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Nader Jafari Rad
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Upper Bounds ◽  
Regular Graphs ◽  
Minimum Cardinality

A subset $S$ of vertices of a graph $G$ is a dominating set of $G$ if every vertex in $V(G)-S$ has a neighbor in $S$. The domination number $\gamma(G)$ of $G$ is the minimum cardinality of a dominating set of $G$. In this paper, we obtain new (probabilistic) upper bounds for the domination number of a graph, and improve previous bounds given by Arnautov (1974), Payan (1975), and Caro and Roditty (1985) for any graph, and Harant, Pruchnewski and Voigt (1999) for regular graphs.

scholarly journals Geodetic Global Domination in the Join of Two Graphs

2020 ◽  
Vol 8 (5) ◽  
pp. 4579-4583
Keyword(s):  
Shortest Path ◽  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Minimum Cardinality ◽  
Dominating Number ◽  
Geodetic Set ◽  
Global Domination Number

A set S of vertices in a connected graph is called a geodetic set if every vertex not in lies on a shortest path between two vertices from . A set of vertices in is called a dominating set of if every vertex not in has at least one neighbor in . A set is called a geodetic global dominating set of if is both geodetic and global dominating set of . The geodetic global dominating number is the minimum cardinality of a geodetic global dominating set in . In this paper we determine the geodetic global domination number of the join of two graphs.

scholarly journals Cycle partitions of regular graphs

2020 ◽  
pp. 1-24
Author(s):  
Vytautas Gruslys ◽  
Shoham Letzter
Keyword(s):  
Regular Graph ◽  
Bipartite Graphs ◽  
Disjoint Paths ◽  
Regular Graphs ◽  
Simple Construction ◽  
Vertex Set ◽  
Almost All ◽  
Vertex Disjoint

Abstract Magnant and Martin conjectured that the vertex set of any d-regular graph G on n vertices can be partitioned into $n / (d+1)$ paths (there exists a simple construction showing that this bound would be best possible). We prove this conjecture when $d = \Omega(n)$ , improving a result of Han, who showed that in this range almost all vertices of G can be covered by $n / (d+1) + 1$ vertex-disjoint paths. In fact our proof gives a partition of V(G) into cycles. We also show that, if $d = \Omega(n)$ and G is bipartite, then V(G) can be partitioned into n/(2d) paths (this bound is tight for bipartite graphs).

scholarly journals Hop Domination in Graphs-II

2015 ◽  
Vol 23 (2) ◽  
pp. 187-199
Author(s):  
C. Natarajan ◽  
S.K. Ayyaswamy
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Total Domination ◽  
Minimum Cardinality ◽  
Unicyclic Graphs ◽  
The Family ◽  
Independent Domination ◽  
Domination In Graphs ◽  
Domination Numbers

Abstract Let G = (V;E) be a graph. A set S ⊂ V (G) is a hop dominating set of G if for every v ∈ V - S, there exists u ∈ S such that d(u; v) = 2. The minimum cardinality of a hop dominating set of G is called a hop domination number of G and is denoted by γh(G). In this paper we characterize the family of trees and unicyclic graphs for which γh(G) = γt(G) and γh(G) = γc(G) where γt(G) and γc(G) are the total domination and connected domination numbers of G respectively. We then present the strong equality of hop domination and hop independent domination numbers for trees. Hop domination numbers of shadow graph and mycielskian graph of graph are also discussed.

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{&#x0394;}(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.

The maximum number of vertices of primitive regular graphs of orders 2, 3, 4 with exponent 2

2021 ◽  
pp. 97-104
Author(s):  
M. B. Abrosimov ◽  
◽  
S. V. Kostin ◽  
I. V. Los ◽  
◽  
...  
Keyword(s):  
Regular Graph ◽  
Regular Graphs ◽  
Similar Question

In 2015, the results were obtained for the maximum number of vertices nk in regular graphs of a given order k with a diameter 2: n2 = 5, n3 = 10, n4 = 15. In this paper, we investigate a similar question about the largest number of vertices npk in a primitive regular graph of order k with exponent 2. All primitive regular graphs with exponent 2, except for the complete one, also have diameter d = 2. The following values were obtained for primitive regular graphs with exponent 2: np2 = 3, np3 = 4, np4 = 11.

A note on improved upper bounds on the transversal number of hypergraphs

2019 ◽  
Vol 11 (01) ◽  
pp. 1950004
Author(s):  
Michael A. Henning ◽  
Nader Jafari Rad
Keyword(s):  
Upper Bound ◽  
Dominating Set ◽  
Domination Number ◽  
Minimum Size ◽  
Total Domination ◽  
Total Domination Number ◽  
Minimum Cardinality ◽  
Total Dominating Set ◽  
Isolated Vertex ◽  
Transversal Number

A subset [Formula: see text] of vertices in a hypergraph [Formula: see text] is a transversal if [Formula: see text] has a nonempty intersection with every edge of [Formula: see text]. The transversal number of [Formula: see text] is the minimum size of a transversal in [Formula: see text]. A subset [Formula: see text] of vertices in a graph [Formula: see text] with no isolated vertex, is a total dominating set if every vertex of [Formula: see text] is adjacent to a vertex of [Formula: see text]. The minimum cardinality of a total dominating set in [Formula: see text] is the total domination number of [Formula: see text]. In this paper, we obtain a new (improved) probabilistic upper bound for the transversal number of a hypergraph, and a new (improved) probabilistic upper bound for the total domination number of a graph.

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