DeepAI AI Chat
Log In Sign Up

Novel ways of enumerating restrained dominating sets of cycles

by   Ratanjeet Pratap Chauhan, et al.

Let G = (V, E) be a graph. A set S ⊆ V is a restrained dominating set (RDS) if every vertex not in S is adjacent to a vertex in S and to a vertex in V - S. The restrained domination number of G, denoted by γ_r(G), is the smallest cardinality of a restrained dominating set of G. Let G_n^i be the family of restrained dominating sets of a graph G of order n with cardinality i, and let d_r(G_n, i)=|G_n^i|. The restrained domination polynomial (RDP) of G_n, D_r(G_n, x) is defined as D_r(G_n, x) = ∑_i=γ_r(G_n)^n d_r(G_n,i)x^i. In this paper, we focus on the RDP of cycles and have, thus, introduced several novel ways to compute d_r(C_n, i), where C_n is a cycle of order n. In the first approach, we use a recursive formula for d_r(C_n,i); while in the other approach, we construct a generating function to compute d_r(C_n,i). We also develop an algorithm, based on integer partitioning and circular permutation, to compute d_r(C_n,i). This gives us an upper bound on the number of restrained dominating sets of a fixed size for C_n.


page 1

page 2

page 3

page 4


[1,2]-Domination in Generalized Petersen Graphs

A vertex subset S of a graph G=(V,E) is a [1,2]-dominating set if each v...

On k-rainbow domination in middle graphs

Let G be a finite simple graph with vertex set V(G) and edge set E(G). A...

On Mixed Domination in Generalized Petersen Graphs

Given a graph G = (V, E), a set S ⊆ V ∪ E of vertices and edges is calle...

Singleton Coalition Graph Chains

Let G be graph with vertex set V and order n=|V|. A coalition in G is a ...

Domination and location in twin-free digraphs

A dominating set D in a digraph is a set of vertices such that every ver...

The Italian domination numbers of some products of directed cycles

An Italian dominating function on a digraph D with vertex set V(D) is de...

On Pseudo-disk Hypergraphs

Let F be a family of pseudo-disks in the plane, and P be a finite subset...