scholarly journals Construction for trees without domination critical vertices

2021 ◽  
Vol 6 (10) ◽  
pp. 10696-10706
Author(s):  
Ying Wang ◽  
◽  
Fan Wang ◽  
Weisheng Zhao ◽  
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Minimum Dominating Set

<abstract><p>Denote by $ \gamma(G) $ the domination number of graph $ G $. A vertex $ v $ of a graph $ G $ is called <italic>fixed</italic> if $ v $ belongs to every minimum dominating set of $ G $, and bad if $ v $ does not belong to any minimum dominating set of $ G $. A vertex $ v $ of $ G $ is called <italic>critical</italic> if $ \gamma(G-v) &lt; \gamma(G) $. By using these notations of vertices, we give a construction for trees that does not contain critical vertices.</p></abstract>

scholarly journals A Note on Total and Paired Domination of Cartesian Product Graphs

10.37236/2535 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
K. Choudhary ◽  
S. Margulies ◽  
I. V. Hicks
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Cartesian Product ◽  
Minimum Dominating Set ◽  
Cartesian Product Of Graphs ◽  
Product Graphs ◽  
Cartesian Product Graphs ◽  
Paired Domination ◽  
Product Of Graphs

A dominating set $D$ for a graph $G$ is a subset of $V(G)$ such that any vertex not in $D$ has at least one neighbor in $D$. The domination number $\gamma(G)$ is the size of a minimum dominating set in G. Vizing's conjecture from 1968 states that for the Cartesian product of graphs $G$ and $H$, $\gamma(G)\gamma(H) \leq \gamma(G \Box H)$, and Clark and Suen (2000) proved that $\gamma(G)\gamma(H) \leq 2 \gamma(G \Box H)$. In this paper, we modify the approach of Clark and Suen to prove a variety of similar bounds related to total and paired domination, and also extend these bounds to the $n$-Cartesian product of graphs $A^1$ through $A^n$.

scholarly journals Computing the Domination Number of Grid Graphs

10.37236/628 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Samu Alanko ◽  
Simon Crevals ◽  
Anton Isopoussu ◽  
Patric Östergård ◽  
Ville Pettersson
Keyword(s):  
Dynamic Programming ◽  
Dominating Set ◽  
Domination Number ◽  
Dynamic Programming Algorithm ◽  
Programming Algorithm ◽  
Dominating Sets ◽  
Grid Graph ◽  
Grid Graphs ◽  
Square Grid ◽  
Minimum Dominating Set

Let $\gamma_{m,n}$ denote the size of a minimum dominating set in the $m \times n$ grid graph. For the square grid graph, exact values for $\gamma_{n,n}$ have earlier been published for $n \leq 19$. By using a dynamic programming algorithm, the values of $\gamma_{m,n}$ for $m,n \leq 29$ are here obtained. Minimum dominating sets for square grid graphs up to size $29 \times 29$ are depicted.

scholarly journals Domination of a Fuzzy Directed Graph

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2143
Author(s):  
Enrico Enriquez ◽  
Grace Estrada ◽  
Carmelita Loquias ◽  
Reuella J. Bacalso ◽  
Lanndon Ocampo
Keyword(s):  
Directed Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Simple Graph ◽  
Fuzzy Graph ◽  
Minimum Dominating Set

A new domination parameter in a fuzzy digraph is proposed to espouse a contribution in the domain of domination in a fuzzy graph and a directed graph. Let GD*=V,A be a directed simple graph, where V is a finite nonempty set and A=x,y:x,y∈V,x≠y. A fuzzy digraph GD=σD,μD is a pair of two functions σD:V→0,1 and μD:A→0,1, such that μDx,y≤σDx∧σDy, where x,y∈V. An edge μDx,y of a fuzzy digraph is called an effective edge if μDx,y=σDx∧σDy. Let x,y∈V. The vertex σDx dominates σDy in GD if μDx,y is an effective edge. Let S⊆V, u∈V\S, and v∈S. A subset σDS⊆σD is a dominating set of GD if, for every σDu∈σD\σDS, there exists σDv∈σDS, such that σDv dominates σDu. The minimum dominating set of a fuzzy digraph GD is called the domination number of a fuzzy digraph and is denoted by γGD. In this paper, the concept of domination in a fuzzy digraph is introduced, the domination number of a fuzzy digraph is characterized, and the domination number of a fuzzy dipath and a fuzzy dicycle is modeled.

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.

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.

Energy-Efficient Sleep Scheduling in WBANs: From the Perspective of Minimum Dominating Set

2019 ◽  
Vol 6 (4) ◽  
pp. 6237-6246 ◽  
Author(s):  
Rongrong Zhang ◽  
Amiya Nayak ◽  
Shurong Zhang ◽  
Jihong Yu
Keyword(s):  
Energy Efficient ◽  
Dominating Set ◽  
Sleep Scheduling ◽  
Minimum Dominating Set
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