scholarly journals A note on the independent domination number versus the domination number in bipartite graphs

2017 ◽  
Vol 67 (2) ◽  
pp. 533-536 ◽  
Author(s):  
Shaohui Wang ◽  
Bing Wei
Keyword(s):  
Domination Number ◽  
Bipartite Graphs ◽  
Independent Domination

scholarly journals Independence and domination on shogiboard graphs

2017 ◽  
Vol 4 (8) ◽  
pp. 25-37 ◽  
Author(s):  
Doug Chatham
Keyword(s):  
Independence Number ◽  
Domination Number ◽  
The Other ◽  
Independent Domination

Abstract Given a (symmetrically-moving) piece from a chesslike game, such as shogi, and an n×n board, we can form a graph with a vertex for each square and an edge between two vertices if the piece can move from one vertex to the other. We consider two pieces from shogi: the dragon king, which moves like a rook and king from chess, and the dragon horse, which moves like a bishop and rook from chess. We show that the independence number for the dragon kings graph equals the independence number for the queens graph. We show that the (independent) domination number of the dragon kings graph is n − 2 for 4 ≤ n ≤ 6 and n − 3 for n ≥ 7. For the dragon horses graph, we show that the independence number is 2n − 3 for n ≥ 5, the domination number is at most n−1 for n ≥ 4, and the independent domination number is at most n for n ≥ 5.

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{Δ}(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.

scholarly journals Independent Domination Stable Trees and Unicyclic Graphs

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 820
Author(s):  
Pu Wu ◽  
Huiqin Jiang ◽  
Sakineh Nazari-Moghaddam ◽  
Seyed Mahmoud Sheikholeslami ◽  
Zehui Shao ◽  
...  
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Minimum Cardinality ◽  
Unicyclic Graphs ◽  
Basic Properties ◽  
Independent Domination ◽  
Independent Dominating Set

A set S ⊆ V ( G ) in a graph G is a dominating set if every vertex of G is either in S or adjacent to a vertex of S . A dominating set S is independent if any pair of vertices in S is not adjacent. The minimum cardinality of an independent dominating set on a graph G is called the independent domination number i ( G ) . A graph G is independent domination stable if the independent domination number of G remains unchanged under the removal of any vertex. In this paper, we study the basic properties of independent domination stable graphs, and we characterize all independent domination stable trees and unicyclic graphs. In addition, we establish bounds on the order of independent domination stable trees.

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