Vol. 2 No. 1 (2018): Vol 2, Iss 1, Year 2018
Articles

Bounds on the Connected Domination in Graphs

Vinolin J
Department of Mathematics, St. Xaviers College, Palayamkottai, Manonmaniam Sundaranar University, Tirunelveli - 627012, Tamil Nadu, India
Ramesh DST
Department of Mathematics, Margchosis College, Nazareth, Manonmaniam Sundaranar University, Tirunelveli - 627012, Tamil Nadu, India
Athisayanathan S
Department of Mathematics, St. Xaviers College, Palayamkottai, Manonmaniam Sundaranar University, Tirunelveli - 627012, Tamil Nadu, India
Anto Kinsley A
Department of Mathematics, St. Xaviers College, Palayamkottai, Manonmaniam Sundaranar University, Tirunelveli - 627012, Tamil Nadu, India
Published June 30, 2018
Keywords
• Graphs, distance, domination number, distance domination number, Hop domination number, Connected Hop domination number.
How to Cite
J, V., DST, R., S, A., & A, A. K. (2018). Bounds on the Connected Domination in Graphs. Journal of Computational Mathematica, 2(1), 1-13. https://doi.org/10.26524/cm22

Abstract

A set S ⊆ V of a connected graph G is a hop dominating set of G if for every vertex v ∈ V S there exists a vertex u ∈ S such that d (u, v) = 2. The cardinality of a minimum hop dominating set of G is called the hop domination number and is denoted by γ_h (G). A hop dominating set D of a graph G is said to be a connected hop dominating set of G if the induced subgraph < D > is connected. The cardinality of a minimum connected hop dominating set is called the connected hop domination number of G and it is denoted by y γ_h^c (G) . In this paper some graphs G are characterized for which γ_h (G) = 2 . Bounds based on diameter, girth and maximum degree for γ_h (G)  are developed. In addition the hop domination number of wounded spider is computed. The hop dominating sets are compared to the distance-2 dominating sets. An important result is proved that if G1,G2,,,,,GS are the connected proper subgraphs of G with minimum connected hop dominating sets D1,D2,,,,,DS as then γ_h^c (G) ≤ γ_h^c (G i) + 2s.