On Edge Dominating Number of Tensor Product of Cycle and Path
Robiatul Adawiyah, Darmaji Darmaji, Reza Ambarwati, Lela Nursafrida, Inge Wiliandani Setya Putri, Ermita Rizki Albirri · International Journal of Advanced Engineering Research and Science · 2017
A subset S' of E(G) is called an edge dominating set ofG if every edge not in S' is adjacent to some edge in S'.The edge dominatingnumber of G, denoted by γ'(G), of G is the minimum cardinality takenover all edge dominating sets of G. Let G1 (V1, E1) and G2(V2,E2) betwo connected graph.The tensor product of G1 and G2, denoted byG1⨂ 𝐺2 is a graph with the cardinality of vertex |V| = |V1| × |V2|and two vertices (u1,u2) and (v1,v2) in V are adjacent in G1⨂ 𝐺2ifu1 v1 ∈ E1 and u2,v2 ∈E2 .In this paper we study an edge dominatingnumber in the tensor product of path and cycle.The results show thatγ'(Cn⨂ 𝑃2) = ⌈ 2𝑛 3 ⌉ for n is odd, γ'(Cn⨂ 𝑃3) = n for n is odd, and theedge dominating number is undefined if n is even.For n ∈even number,we investigated the edge dominating number of its component on tensorproduct of cycle Cn and path.The results are γ'c(Cn⨂ 𝑃2)= ⌈ 𝑛 3 ⌉ andγ'c(Cn ⨂ 𝑃3) = ⌈ 𝑛 2 ⌉ which Cn ,P2 and P3, respectively, is Cycle order n,Path order 2 and Path order 3.