Research Article Reinforcement Number of a Graph with respect to Half-Domination G. Muhiuddin , 1 N. Sridharan, 2 D. Al-Kadi, 3 S.Amutha , 4 andM.E.Elnair 1,5 1 Department of Mathematics, University of Tabuk, Tabuk 71491, Saudi Arabia 2 Department of Mathematics, Alagappa University, Karaikudi 630 003, India 3 Department of Mathematics and Statistics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia 4 Ramanujan Centre for Higher Mathematics, Alagappa University, Karaikudi 630 003, India 5 Department of Mathematics and Physics, Faculty of Education, Hasahesa, Gezira University, Sudan Correspondence should be addressed to S. Amutha; amuthas@alagappauniversity.ac.in Received 17 December 2020; Revised 1 February 2021; Accepted 30 March 2021; Published 14 April 2021 Academic Editor: Ismail Naci Cangul Copyright©2021G.Muhiuddinetal.isisanopenaccessarticledistributedundertheCreativeCommonsAttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this paper, we introduce the concept of reinforcement number with respect to half-domination and initiate a study on this parameter. Furthermore, we obtain various upper bounds for this parameter. AMS subject classification: 05C38, 05C40, 05C69. 1.Introduction roughout our discussion, we consider only simple finite graphs. For graph theoretic terminologies, we refer the readers to [1]. If G �(V, E) is a graph to each vertex v of G, N 1 (v) denotes the set of all vertices of G which are adjacent to v. For any subset S⊆V, N 1 (S)� ∪ N 1 (v): v ∈ S  . For a vertex v of G, N 1 [v]� N 1 (v) ∪ v {} and N 2 [v]� u ∈ V: d { (u, v)� 2}. A vertex v ∈ V is said to dominate itself and its adjacent vertices. In other words, a vertex v dominates a vertex u if u ∈ N 1 [v]. A subset D of V(G) is said to be a dominating set of G if V � ∪ v∈D N 1 [v]. e minimum cardinality of a dominating set D of G is denoted by c(G) and is called the domination number of G. e total domination number of a graph G, denoted by c t (G), is the minimum cardinality of a total dominating set of G and, for their properties, we refer the reader to [2–6]. In [7], Kulli introduced the concept of the cobondage number cb(G) ofa graph, which is the minimum number of edges to be added to reduce the domination number. e same concept has been independently introduced and studied earlier by others under the name “reinforcement number” (refer to chapter 17 of Haynes et al.’s work [8]). Total reinforcement number of a graph has been studied in [9–12]. Recently, Muhiuddin et al. have studied various related concepts on graphs (see, e.g., [13–17]). In [18, 19], a new domination parameter c λ (G), where 0 < λ < 1, was introduced and a study on c (1/2) (G) had been initiated. In this paper, we introduce the concept of rein- forcement number with respect to half-domination and initiate a study on this parameter. 2.DefinitionandExamples If G is a graph for every vertix u of G, we define a map f u : V(G) ⟶ 0, (1/2), 1 { } as follows: f u (v)� 1, ifd(u, v) ≤ 1, 1 2 , ifd(v, u)� 2, 0, otherwise, ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (1) for all v ∈ V(G). e map f u is called the half-domination factor of u. A subset D of V(G) is said to be a (1/2)-dominating set for G if, for each v ∈ V(G),  u∈D f u (v) ≥ 1. e minimum Hindawi Journal of Mathematics Volume 2021, Article ID 6689816, 7 pages https://doi.org/10.1155/2021/6689816