Research Article Bounds on Co-Independent Liar’s Domination in Graphs K.SuriyaPrabha , 1 S.Amutha , 2 N. Anbazhagan , 1 andIsmailNaciCangul 3 1 Department of Mathematics, Alagappa University, Karaikudi-630 003, Tamilnadu, India 2 Ramanujan Centre for Higher Mathematics (RCHM), Alagappa University, Karaikudi-630003, Tamilnadu, India 3 Department of Mathematics, Bursa Uludag University, Gorukle 16059, Turkey Correspondence should be addressed to S. Amutha; amuthas@alagappauniversity.ac.in Received 16 January 2021; Revised 22 February 2021; Accepted 3 March 2021; Published 20 March 2021 Academic Editor: Ghulam Shabbir Copyright © 2021 K. Suriya Prabha et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A set S⊆V of a graph G �(V, E) is called a co-independent liar’s dominating set of G if (i) for all v ∈ V, |N G [v] ∩ S| ≥ 2, (ii) for every pair u, v ∈ V of distinct vertices, |(N G [u] ∪ N G [v]) ∩ S| ≥ 3, and (iii) the induced subgraph of G on V − S has no edge. e minimum cardinality of vertices in such a set is called the co-independent liar’s domination number of G, and it is denoted by c LR coi (G). In this paper, we introduce the concept of co-independent liar’s domination number of the middle graph of some standard graphs such as path and cycle graphs, and we propose some bounds on this new parameter. 1.Introduction For notations and nomenclature, we refer [1]. Specifically, let G �(V, E) be a graph with vertex set V of order p �|V| and edge set E of size q �|E|. e diameter of G is the greatest distance between any two vertices of G. e middle graph M(G) is the derived graph obtained from G byinsertinganew vertex into every edge of G and then joining these new vertices by edges which lie on the adjacent edges of G [2]. Haynes et al. introduced the concept of domination in graphs [3]. A topological index is a real number related to a graph, which must be a structural invariant. e topological indices are a vital tool for quantitative structure activity relationship and quantitative structure property relationship. For more work on topological indices of a graph, refer recent papers [4, 5]. e concept of liar’s domination was introduced by Panda and Paul in [6]. A graph G �(V, E) admits a liar’s dominating set if each of its connected components has at least three vertices. Several different domination parameters were studied in [7–12]. For references on liar’s domination, see, for instance, [2, 13]. A subset S⊆V of a graph G �(V, E) is called a co-independent liar’s dominating set of G if (i) for all v ∈ V, |N G [v] ∩ S| ≥ 2, (ii) for every pair u, v ∈ V of distinct vertices, |(N G [u] ∪ N G [v]) ∩ S| ≥ 3, and (iii) the induced subgraph of G on V − S has no edge. e minimum cardinality of vertices in such a set is called the co-independent liar’s domination number of G, and it is denoted by c LR coi (G). In this paper, we initiate the study of co-independent liar’s domination in graphs. 2.Co-IndependentLiar’s DominationinGraphs In this section, we first strengthen the co-independent liar’s domination number of the middle graphs of some standard graphs. Eventually, some bounds will be obtained. Theorem 1. Let M(P p ) be the middle graph of a path graph P p of order p. en, c LR coi MP p     ≤ p + 1. (1) Proof. Let u 1 ,u 2 ,u 3 , ... ,u p betheverticesof P p andalsothe vertices in V(M(P p )− P p ) be u p+1 ,u p+2 ,u p+3 , ... ,u 2p− 1 .Let u ∈ V(M(P p )).Weprovethatalltheverticesof M(P p ) geta co-independent liars dominating set arising in four cases: (1) Case(i):let u � u 1 .Recallthatdeg(u 1 )� deg(u p )� 1. So, N[u 1 ]� u 1 ,u p+1   in M(P p ) and |N[u 1 ]| � 2. erefore, |N[u] ∩ S| ≥ 2, for all u ∈ V(M(P p )), and Hindawi Journal of Mathematics Volume 2021, Article ID 5544559, 6 pages https://doi.org/10.1155/2021/5544559