Recoletos Multidisciplinary Research Journal (Dec 2016)

Perfect Outer-connected Domination in the Join and Corona of Graphs

  • Enrico Enriquez,
  • Valerie Fernandez,
  • Teodora Punzalan,
  • Jonecis Dayap

DOI
https://doi.org/10.32871/rmrj1604.02.01
Journal volume & issue
Vol. 4, no. 2

Abstract

Read online

Let 𝐺 be a connected simple graph. A dominating set 𝑆 βŠ† 𝑉(𝐺) is called a perfect dominating set of 𝐺 if each 𝑒 ∈ 𝑉 𝐺 βˆ– 𝑆 is dominated by exactly one element of 𝑆. A set 𝑆 of vertices of a graph 𝐺 is an outer-connected dominating set if every vertex not in 𝑆 is adjacent to some vertex in 𝑆 and the subgraph induced by 𝑉 𝐺 βˆ– 𝑆 is connected. A perfect dominating set 𝑆 of a graph 𝐺 is a perfect outer-connected dominating set if the subgraph induced by 𝑉 𝐺 βˆ– 𝑆 is connected. The perfect outer-connected domination number of 𝐺, denoted by 𝛾 𝑐 𝑝 (𝐺), is the smallest cardinality of a perfect outer-connected dominating set 𝑆 of 𝐺. A perfect outer-connected dominating set with cardinality 𝛾 𝑐 𝑝 (𝐺) is called 𝛾 𝑐 𝑝 -𝑠𝑒𝑑 of 𝐺. In this paper, we will show that given positive integers, π‘Ž, 𝑏, 𝑐, and 𝑛 such that π‘Ž ≀ 𝑏 ≀ 𝑐 ≀ 𝑛 βˆ’ 1, there exists a connected graph 𝐺 with |𝑉(𝐺)| = 𝑛, 𝛾(𝐺) = π‘Ž, 𝛾𝑝(𝐺) = 𝑏, and 𝛾 𝑐 𝑝 𝐺 = 𝑐. Further, we give the characterization of the perfect outer-connected dominating set of the join and corona of two graphs and give their corresponding perfect outer-connected domination number.

Keywords