On the Saturation Number of Graph Powers

Let 𝐺 = (𝑉, 𝐸) be a simple connected graph. A matching of G is a set of disjoint edges of G. A matching 𝑀 in 𝐺 is maximal if no other matching of 𝐺 has 𝑀 as a proper subset. The saturation number of 𝐺 is the cardinality of any smallest maximal matching in 𝐺 and is denoted by 𝑠(𝐺). For every positive integer 𝑛, the 𝑛-subdivision of G is a simple graph 𝐺 1 𝑛 which is constructed by replacing each edge of 𝐺 with a path of length 𝑛. The 𝑚th power of 𝐺, denoted by 𝐺𝑚, is a graph with the same vertex set as 𝐺 such that two vertices are adjacent in 𝐺𝑚 if and only if their distance is at most 𝑚 in 𝐺. The 𝑚th power of the 𝑛-subdivision of 𝐺 has been introduced as a fractional power of 𝐺 and is denoted by 𝐺 𝑚 𝑛 . In this paper we study the saturation number of the natural and the fractional powers of some graphs.