New Counting Formula for Dominating Sets in Path and Cycle Graphs

Abstract

Let G=(V(G), E(G)) be a path or cycle graph. A subset D of V(G) is a dominating set of G if for every u element of V(G)\D, there exists v element of D such that uv element of E(G), that is, N[D]=V(G). The domination number of G, denoted by gamma(G), is the smallest cardinality of a dominating set of G. A set D_1 subset of V(G) is a set containing dominating vertices of degree 2, that is, each vertex is internally stable. A set D_2 subset of V(G) is a set containing dominating vertices where one of the element say a element of D_2, and the rest are of degree 2. A set D_3 subset of V(G) is a set containing dominating vertices in which two of the elements say b, c element of D_3, deg(b)=deg(c)=1. This paper developed a new combinatorial formula that determines the number of ways of putting a dominating set in a path and cycle graphs of order n>=1 and n>=3, respectively. Further, a combinatorial function P^1_G(n), P^2_G(n) and P^3_G(n) that determines the probability of getting the set D_1, D_2, and D_3, respectively in graph G of order n were constructed.

A Closer Look at a Path Domination Number in Grid Graphs

“…(𝐺) and 𝑁 𝛾 𝑝𝑎𝑡ℎ (𝐺) be the number of ways of putting a k-path dominating set and path dominating set in graph 𝐺, respectively. For the readers, the number of ways of putting the dominating set in the path and cycle graph can be read in [18]. Then, the following result is quick from Remark 2.1 and Theorem 2.3.…”

A Note on Triple Repetition Sequence of Domination Number in Graphs

“…The proof directly follows from Theorem 2. In the paper of Casinillo [17], domination in path and cycle graphs was revisited, and investigated some undiscovered properties. The author has developed a new formula that determines the number of ways in putting a dominating set of vertices and provided a combinatorial proof.…”

“…Theorem 2.5. [17] Let 𝐺 = 𝑃 𝑛 where 𝑛 is a positive integer. Then, In that case, we can extend the result by applying it to a sequence of paths with consecutive orders.…”