Abstract
For graph G = (V, E), the open neighborhood of a vertex v is N(v) = {u ∈ V|uv ∈ E}. A function f : V (G) → {0, 1, 2} is called an Italian dominating function of G if Σ u∈N(v) f (u) ≥ 2, for each vertex f (v) = 0. The weight of f is w( f) = Σ v∈V(G) f (v). The minimum weight of an Italian dominating function of G is called the Italian domination number of G, denoted by y I (G). In this paper, we present a bagging approach and a partitioning approach to investigate the Italian domination number of Cartesian product of circles and paths, Cn□ P m . We determine the exact values of the Italian domination numbers of C n □ P 3 , C 3 □ P m . We also present some bounds on the Italian domination number of C n □ P m for n, m ≥ 4.
Highlights
We first present some necessary terminology and notation
We determine the exact values of the Italian domination number of Cn P3 and C3 Pm
In the preceding section, we present a bagging approach which we use on domination type problem. we use this approach for Italian domination on C3 Pm and Cn P3
Summary
We first present some necessary terminology and notation. Let G be a finite, simple, and undirected graph with vertex set V (G) and edge set E(G). For Roman domination, each vertex in the graph corresponds to a location in the Roman Empire. A Roman dominating function of weight γR(G) corresponds to such an optimal assignment of legions to locations. The minimum weight of an RDF on G is called the Roman domination number of G, denoted by γR(G). A function f : V (G) → {0, 1, 2} is called an Italian dominating function (IDF) on G, if u∈N(v) f (u)≥2, for each vertex f (v) = 0. The minimum weight of an IDF on G is called the Italian domination number of G, denoted by γI (G). We present a bagging approach and a partitioning approach to investigate the Italian domination number of Cartesian product of circles and paths.
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