Total Neighborhood Number of a Graph

**ABSTRACT:**

A set S of vertices of a graph G is a total neighborhood set of G if G is the union of the subgraphs induced by the closed neighborhoods of the vertices in S and for every vertex uV there exists a vertex v S such that u is adjacent to v. The total neighborhood number nt(G) of G is the minimum cardinality of a total neighborhood set of G. A total neighborhood nomatic partition of G is a partition {V1, V2, …, Vk} of V in which each Vi is a total neighborhood set of G. The total neighborhood nomatic number ntn(G) of G is the maximum order of a partition of the vertex set of G into total neighborhood sets. In this paper, we obtain results about two parameters, the total neighborhood number and total

neighborhood nomatic number.

**KEYWORDS: **graph, total neighborhood number, total neighborhood nomatic number. Mathematics Subject Classification: 05C

**DOI:**jusps-A

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A set S of vertices of a graph G is a total neighborhood set of G if G is the union of the subgraphs induced by the closed neighborhoods of the...