EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 5918 ISSN 1307-5543 – ejpam.com Published by New York Business Global Internally-Locating Dominating Sets in Graphs Irish S. Tropico1,∗, Isagani S. Cabahug, Jr.1 1 Department of Mathematics, College of Arts and Sciences, Central Mindanao University, Musuan, Maramag, Bukidnon, 8714 Philippines Abstract. For a connected graph G, a subset I ⊆ V (G) is a locating-dominating set if it is a dominating set and for every two distinct vertices x, y ∈ V (G) \ I, N(x) ∩ I ̸= N(y) ∩ I. This paper introduces the concept of an internally-locating dominating set. Specifically, a nonempty set I ⊆ V (G) with |I| ≥ 2 is an internally-locating set in a nontrivial connected graph G if and only if, for every u, v ∈ I, N(u)∩ I ̸= N(v)∩ I. Thus, I is an internally-locating dominating set if it is both an internally-locating set and a dominating set. In addition, this paper identifies some properties of this concept, provides characterizations of certain special classes of graphs, including total graphs and shadow graphs with ∆(G) = 2, with their corresponding internally-locating domination number, and cases where γli(G) = 2. Moreover, it provides a sufficient condition for γ(G) = γli(G), specifically when G is a corona product. 2020 Mathematics Subject Classifications: 05C69 Key Words and Phrases: Internally-locating set, internally-locating dominating set, internally- locating domination number, total graph, shadow graph, corona of graph 1. Introduction Over time, graph theory has evolved into a powerful tool for analyzing networks and solving optimization problems. Two fundamental concepts in this field are dominating sets and locating sets. Domination theory, introduced by Berge and Ore in the 1960s [1], has its applications in various areas, including facility location problems (FLPs) [2]. FLPs focus on optimizing factors such as transportation costs and market share [2]. Locating sets, introduced by Slater [3], are also important, with applications in systems like sonar and long-range navigation (LORAN) stations [4]. In 1998, Slater combined the ideas of domination and location to introduce locating- dominating sets [5], which help identify vertices while also dominating the graph. These sets have applications in fields such as fire location detection and multiprocessor error diagnosis [6]. Slater’s work, along with subsequent studies by Canoy and Omega, expanded ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5918 Email addresses: irish.tropico17@gmail.com (I. Tropico), isaganicabahugjr@cmu.edu.ph (I. Cabahug, Jr.) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) I. Tropico, I. Cabahug, Jr. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5918 2 of 21 the theory by exploring locating sets in complex graph operations, such as the join and corona of graphs [7]. Canoy and Malacas further characterized locating-dominating sets in graph compositions and corona graphs [8]. Although much progress has been made in understanding dominating and locating sets, there are still unexplored concepts. This led to a question about the distinctness within the set. This study introduces internally-locating dominating sets and explores their properties in special graphs, particularly focusing on unary operations such as total and shadow graphs with ∆(G) = 2, along with their internally-locating domination numbers, and binary operations such as the corona of graphs. Some aspects of the proofs in this study were inspired by [9]. 2. Terminology and Notation This study considers finite, simple, nontrivial, connected, and undirected graphs. Some definitions of the concepts covered in this study is included within. For fundamental graph- theoretic concepts and additional terms, the readers may refer to [10], [11], [4], [12], [13], [14], [7], [15], [16]. Let G = (V,E) be a graph. If the pair u and v is in E, then e = uv is an edge of G and the vertices u and v are adjacent in G. Two adjacent vertices in G are referred to as neighbors of each other. The set of neighbors of a vertex v is called the open neighborhood of v denoted by NG(v) = N(v) [4]. The degree of a vertex v in a graph G is the number of vertices that are adjacent to v, denoted by degG(v) or simply deg(v). A vertex of degree 0 is referred to as an isolated vertex and a vertex of degree 1 is an end-vertex or leaf. The largest degree among the vertices of G is called the maximum degree of G, denoted by ∆(G) [4]. A set S ⊆ V (G) is a dominating set of G, if every vertex in V (G) \ S is adjacent to at least one vertex in S. The domination number γ(G) is the minimum cardinality of dominating set [10]. A subset S of V (G) is a locating set in a connected graph G if for any two distinct vertices u and v in V (G) \ S, NG(u) ∩ S ̸= NG(v) ∩ S [7]. A set D ⊆ V is a locating-dominating set if it is dominating and every two vertices x, y ∈ V (G) \ S,N(x) ∩ D ̸= N(y) ∩ D. The locating-domination number, denoted by γL(G) is the minimum cardinality of an locating-dominating set of G [12]. A gear graph, denoted by Gn, is obtained from the wheel graph by adding a vertex between every pair of adjacent vertices of the cycle [16]. Example 1. Consider the wheel graph W4 with V (W4) = {u}∪{v1, v2, v3, v4} and E(W4) = {uvi|1 ≤ i ≤ 4} ∪ {v1v2, v2v3, v3, v4, v1v4}. The gear graph G4 in Figure 1, is obtained by adding vertices a0, a1, a2, and a3 between the vertices v1 and v2, v2 and v3, v3 and v4, and v4 and v1, respectively. I. Tropico, I. Cabahug, Jr. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5918 3 of 21 v1 v3 v2 v4 u a0 a2 a1 a3 Figure 1: The gear graph G4 An banana tree graph is a graph obtained by connecting one leaf to each of n copies of a star graph K1,k with a single root vertex that is distinct from all the stars, denoted by Bn,k+1 [17]. Example 2. Consider the star graph K1,3 with V (K1,3) = {x1, y1, y2, y3}. Then in Figure 2 is a banana tree graph B3,4 obtained by connecting one leaf of three copies of star K1,3 with a single root vertex u. x1,1 x1,3x1,2 u y2,1 y1,1 y3,1 y2,2 y1,2 y3,2 y2,3 y1,3 y3,3 Figure 2: The banana tree graph B3,4 A lollipop graph is the graph obtained by joining a complete graph Km to path graph Pn with a bridge, denoted by Lm,n [17]. Example 3. Consider the complete graph K3 with V (K3) = {v1, v2, v3} and a path graph P4 with V (P4) = {x1, x2, x3, x4}. Then in Figure 3 is a lollipop graph L3,4 obtained from joining the vertex v3 and x1. v1 v2 v3 x1 x2 x3 x4 Figure 3: The lollipop graph L3,4 I. Tropico, I. Cabahug, Jr. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5918 4 of 21 The total graph T (G) of a graph G, is a graph G such that the vertices set of T (G) corresponds to the vertices and edges of G and two vertices are adjacent in T (G) if and only if their corresponding elements are either adjacent or incident in G. The total graph has a vertex set V (T (G)) = V (G) ∪ E(G) [18]. Example 4. Consider the path graph P4, V (P4) = {v1, v2, v3, v4} and E(P4) = {e1, e2, e3} such that e1 = v1v2, e2 = v2v3, and e3 = v3v4. Then in Figure 4 is the total graph of path graph P4, denoted by T (P4) has V (T (P4)) = {v1, v2, v3, v4, e1, e2, e3} with each of the vertices in the vertex set of T (P4) is joined by an edge if it is adjacent or incident in P4. v1 v2 v3 v4 e1 e2 e3 Figure 4: The total graph of path graph P4 The shadow graph S(G) of a graph G is obtained from G by adding, for each vertex v of G, a new vertex v′, called the shadow vertex of v, and joining v′ to the neighbors of v in G. The set of all shadow vertices is denoted by V ′(G) [4]. Example 5. Consider the path graph P4 in Figure 4 with V (P4) = {v1, v2, v3, v4}. Then the shadow graph of path graph P4 is shown in Figure 5 with V ′(P4) = {v′1, v′2, v′3, v′4}. v1 v2 v3 v4 v′1 v′2 v′3 v′4 Figure 5: The shadow graph of path graph P4 3. Known Result Corollary 1. [19] Let G be a connected graph of order m, and let H be any graph of order n. Then, γ(G ◦H) = m, where γ(G ◦H) represents the domination number of the corona product G ◦H. I. Tropico, I. Cabahug, Jr. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5918 5 of 21 4. Results This paper uses the following terms to denote specific concepts: degI(v) represents the degree of vertex v within the set I, NI(v) signifies the neighborhood of vertex v within the set I, γ − set refers to the minimum dominating set, ILS signifies an internally-locating set, ILDS denotes an internally-locating dominating set, and the γli − set represents the minimum internally-locating dominating set. Definition 1. A non-empty set S ⊆ V (G) with |S| ≥ 2 is an internally-locating set in a connected graph G if and only if for every u, v ∈ S, the N(u) ∩ S ̸= N(v) ∩ S. Example 6. Consider the graph G in Figure 6, and a set I = {v2, v3, v5} ⊂ V (G). Note that N(v2) ∩ I = {v3, v5}, N(v3) ∩ I = {v2, v5}, and N(v5) ∩ I = {v2, v3}. By Definition 1, I is an internally-locating set in G. v1 v2 v3 v4 v5 v6 Figure 6: A graph G Definition 2. A set I ⊆ V (G) such that |I| ≥ 2 is an internally-locating dominating set if and only if I is a dominating set and an internally-locating set in G. Moreover, the minimum cardinality of internally-locating dominating set, denoted by γli(G) is called an internally-locating domination number of G. Example 7. Consider the graph G in Figure 6, and a set I = {v2, v5} ⊂ V (G). Note that N(v2)∩ I = {v5} ≠ {v2} = N(v5)∩ I. By Definition 1, I is an internally-locating set in G. Note that I is a dominating set in G and also a γ − set. By Definition 2, I is an internally-locating dominating set and also a γli − set in G. Therefore, γli(G) = 2. Theorem 1. Let G be a graph and I ⊆ V (G) such that |I| ≥ 2 and I is a dominating set. Then I is not an internally-locating dominating set in G if one of the following holds: (i) H is a component of G[I] such that H = P3 or C4; (ii) I induced more than one isolated vertex or an empty graph of order n ≥ 2; (iii) the component of G[I] = Km,n, where m ≥ 1 and n ≥ 3; or (iv) the component of G[I] = D4. I. Tropico, I. Cabahug, Jr. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5918 6 of 21 Proof. Let G be a graph and I ⊆ V (G) such that |I| ≥ 2. Assume that (i)− (iv) hold. If H = P3, consider V (H) = V (P3) = {v1, v2, v3} and E(H) = E(P3) = {v1v2, v2v3}. Note that H ⊆ G[I], thus V (H) ⊆ I. Observe that N(v1) ∩ I = {v2} = N(v3) ∩ I. Thus, I is neither an ILS nor ILDS. Similarly, if H = C4, let V (H) = V (C4) = {c1, c2, c3, c4} and E(H) = E(C4) = {c1c2, c2c3, c3c4, c4c1}. Note that H ⊆ G[I], thus V (H) ⊆ I. Observe that N(c1) ∩ I = {c2, c4} = N(c3) ∩ I. Thus, I is neither an ILS nor ILDS. Now, if I induces an isolated vertex v, then N(v) ∩ I = ∅. By (ii), there exists u ∈ I, u ̸= v, such that N(u) ∩ I = ∅. Thus, N(v) ∩ I = ∅ = N(u) ∩ I. Hence, I is neither an ILS nor ILDS. By the definition of an empty graph, this generally holds if I induces an empty graph of order n ≥ 2. By (iii), let H = Km,n, m ≥ 1 and n ≥ 3. By the definition of Km,n, V (H) = V (Km,n) = {x1, x2, . . . , xm}∪ {y1, y2, . . . , yn} and E(Km,n) = {xiyj |1 ≤ i ≤ m, 1 ≤ j ≤ n}. Note that H ⊆ G[I], thus V (H) ⊆ I. WLOG, consider x1 and x2 ∈ V (Km,n). This implies N(x1)∩I = {y1, y2, . . . , yn}, and N(x2)∩I = {y1, y2, . . . , yn}. Thus, N(x1)∩I = N(x2)∩I. Hence, I is neither an ILS nor ILDS. This is also true for H = K1,n, n ≥ 3. Finally, by (iv), if H = D4, let V (H) = V (D4) = {d1, d2, d3, d4} and E(H) = E(D4) = {d1d2, d2d3, d3d4, d4d1, d1d3}. Note that H ⊆ G[I], thus V (H) ⊆ I. Observe that N(d2)∩ I = {d1, d3} = N(d4) ∩ I. Thus, I is neither an ILS nor ILDS. Theorem 2. Let G be a graph. Then γli(G) = 2 if one of the following hold: (i) γ(G) = 1 (ii) γ(G) = 2, where {u, v} is the γ − set and u, v are adjacent. Proof. Let G be a graph. Case 1: If γ(G) = 1, this means that there exists a single vertex v ∈ V (G) such that N(v) = V (G) \ {v}. In this case, {v} is the γ-set. This implies that for any u ∈ V (G), u ̸= v, {u, v} is a dominating set. Since N(v) = V (G) \ {v}, and u is dominated by v, this implies N(u) ∩ {u, v} = {v} and N(v) ∩ {u, v} = {u}. Thus, N(u) ∩ {u, v} ̸= N(v) ∩ {u, v}, which confirms that {u, v} is an internally-locating set. By definition of γli − set, this follows that γli(G) = 2 in this case. Case 2: If γ(G) = 2, where {u, v} is the γ − set and u, v are adjacent. Clearly, {u, v} is γli − set. Hence, γli(G) = 2 in this case. Corollary 2. The γli(G) = 2 if (i) G = Pn = Cn, 2 ≤ n ≤ 3; (ii) G = Kn, n ≥ 2 I. Tropico, I. Cabahug, Jr. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5918 7 of 21 (iii) G = K1,n, n ≥ 2; (iv) G = Fn, n ≥ 2; and (v) G = Wn, n ≥ 4. Proof. This follows from the definition of dominating set and Theorem 2 (i). Theorem 3. Let G be a graph with n ≥ 4. Then γ(G) = γli(G) if G = Pn ◦H, for any graph variation of H. Proof. Let G be a graph with n ≥ 4 and G = Pn ◦ H, for any graph variation of H. By Corollary 1, γ(G) = n. By definition of path graph Pn and since G = Pn ◦ H, V (Pn) = {v1, v2, . . . , vn} is a γ − set. Now, observe that N(v1) ∩ V (Pn) = {v2}, N(vi) ∩ V (Pn) = {vi−1, vi+1}, 2 ≤ i ≤ n− 1 N(vn) ∩ V (Pn) = {vn−1}. Thus, V (Pn) is an ILS and γli − set. Hence γli(G) = n. Therefore, γ(G) = γli(G). Theorem 4. Let G be a graph with ∆(G) = 2 and I ⊆ V (G) such that |I| ≥ 2. Then I is the minimum internally-locating dominating set in G if and only if the following holds: (i) if degI(v) = 1, then degV (G)\I(v) = 1; (ii) if degI(v) = 0, then v is unique in I; (iii) if degI(v) = 0 and deg(v) = 1 or 2, then degV (G)\I(v) = deg(v); and (iv) for all u ∈ V (G) \ I, u ∈ N(v) for some v ∈ I. Proof. Let G be a graph with ∆(G) = 2 and I ⊆ V (G) such that |I| ≥ 2. Assume I is the γli− set in G. Now, if degI(v) = 1, v has one neighbor in I. Since I is dominating and ∆(G) = 2, v must have one neighbor in V (G) \ I, i.e., degV (G)\I(v) = 1. Thus, (i) holds. For (ii), since I is an ILDS, we have N(v)∩I = ∅ if degI(v) = 0. Suppose that v is not unique in I, this implies that there exists u ∈ I where u ̸= v such that degI(u) = 0. Thus, N(u) ∩ I = ∅ = N(v) ∩ I. A contradiction, since I is an ILS. Thus, v with degI(v) = 0 is unique in I. If degI(v) = 0 and deg(v) = 1 or 2, degV (G)\I(v) = deg(v), since all neighbors of v must lie in V (G) \ I. Hence, (iii) holds. Lastly, by definition of dominating set, then for all u ∈ V (G) \ I, u is adjacent to at least one v ∈ I, as I is dominating. Thus, (iv) holds. I. Tropico, I. Cabahug, Jr. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5918 8 of 21 Conversely, assume I satisfies (i)-(iv). By (iv), clearly, I is a dominating set. By (i), it follows that for every v ∈ I, there exist a unique u ∈ I such that N(v) ∩ I = u. By (ii) and (iii), there exist no u, v ∈ I such that N(u) ∩ I = ∅ = N(v) ∩ I. Thus, I is an ILS. Now, by removing any vertex in I, then it contradicts (iv). Hence, I is the γli − set. Corollary 3. Let G be a path graph Pn or a cycle graph Cn with n ≥ 4. Then, γli(G) = γli(Pn) = γli(Cn) =  n 2 if n ≡ 0, 2 (mod 4) n+1 2 if n ≡ 1 (mod 4) n−1 2 if n ≡ 3 (mod 4) Proof. Let G be a path graph Pn or a cycle graph Cn with n ≥ 4. For convenience, let V (Pn) = V (Cn) = {v1, v2, v3, . . . , vn−1, vn}, E(Pn) = {v1v2, v2v3, . . . , vn−1vn} and E(Cn) = {v1v2, v2v3, . . . , vn−1vn, vnv1}. If G = Pn, consider the following cases: Case 1: n ≡ 0 (mod 4) Let I ⊆ V (Pn) with I = {v2, v3, v6, v7, . . . , vn−2, vn−1}. Observe that |I| = n 2 . By Theorem 4, I is a γli − set. Therefore, γli(Pn) = |I| = n 2 . Case 2: n ≡ 1 (mod 4) Let I ⊆ V (Pn) with I = {v2, v3, v6, v7, . . . , vn−3, vn−2, vn}. Observe that |I| = n+1 2 . By Theorem 4, I is a γli − set. Therefore, γli(Pn) = |I| = n+1 2 . Case 3: n ≡ 2 (mod 4) Let I ⊆ V (Pn) with I = {v2, v3, v6, v7, . . . , vn−2, vn−1}. Observe that |I| = n 2 . By Theorem 4, I is a γli − set. Therefore, γli(Pn) = |I| = n 2 . Case 4: n ≡ 3 (mod 4) Let I ⊆ V (Pn) with I = {v2, v3, v6, v7, . . . , vn−5, vn−4, vn−1}. Observe that |I| = n−1 2 . By Theorem 4, I is a γli − set. Therefore, γli(Pn) = |I| = n−1 2 . A similar argument follows for G = Cn, as the number of vertices is the same as in the path graph Pn, resulting the same values for γli(G) in all cases. Theorem 5. Let G be a banana tree graph Bn,k+1 with n ≥ 2 and k ≥ 3, and I ⊆ V (G) with |I| ≥ 2. Then, I is the minimum internally-locating dominating set of G if and only if the following holds: (i) for every ui ∈ V (K1i,ki), then ui ∈ I; (ii) there exists exactly one ui ∈ V (K1i,ki) such that degI(ui) = 0; I. Tropico, I. Cabahug, Jr. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5918 9 of 21 (iii) if degI(ui) ̸= 0, then degI(uj) = 1, i ̸= j; and (iv) there exist v ∈ I, such that r ∈ N(v). Where ui ∈ V (K1i,ki) is the central vertex of the ith copy of a star graph K1i,ki, 1 ≤ i ≤ n and r a single root vertex of G. Proof. Let G = Bn,k+1 be a banana tree graph, where n ≥ 2 and k ≥ 3, and let I ⊆ V (G) with |I| ≥ 2. Suppose that ui ∈ V (K1i,ki) be a central vertex of the ith copy of a star graph K1i,ki , 1 ≤ i ≤ n and r a single root vertex of G. Assume I is the γli − set of G. Then, I is a dominating set, so every vertex in G must be adjacent to at least one vertex in I. In K1i,ki , the central vertex ui is connected to all ki leaves. To dominate the leaves of K1i,ki , ui must belong to I. Thus, (i) holds. By definition of ILDS, for any u, v ∈ I, their neighborhoods within I must satisfy N(u)∩ I ̸= N(v)∩ I. To minimize |I|, exactly one ui in V (K1i,k) must have degI(ui) = 0. Hence, (ii) holds. Now, if degI(ui) = 0, since (ii) holds, degI(uj) ≥ 1, i ̸= j. Then, by Theorem 1 (i) and (iii), this follows degI(uj) ≤ 1. Thus, degI(uj) = 1, i ̸= j. Lastly, by I as an ILDS, and since (iii) holds, to dominate r, there must exist v ∈ I such that r ∈ N(v). Conversely, assume that I ⊆ V (G) satisfies conditions (i)− (iv). By (i) and (iv), every vertex in G is dominated by I. By (ii), there exists ui ∈ I such that NI(ui) = ∅, and by (iii), for all uj ∈ I, i ̸= j, NI(uj) is a singleton set. Since degI(uj) = 1, i ̸= j, it follows that uj is adjacent to one of the leaves of the star graph K1j ,kj . Note that for all vi ∈ V (K1i,ki), N(vi) ̸= vj , for all vj ∈ V (K1j ,kj ), i ̸= j. Consequently, for all u, v ∈ I, N(u) ∩ I ̸= N(v) ∩ I. Thus, I is an ILDS. Now, by removing any vertex v ∈ I, it violates either domination or ILS property. Hence, I is a γli − set. Corollary 4. Let G be a banana tree graph Bn,k+1 with n ≥ 2 and k ≥ 3. Then, γli(G) = 2n− 1. Proof. By Theorem 5 (i)-(iii), |I| = 2n− 1. Hence, γli(G) = 2n− 1. Theorem 6. Let G be a gear graph Gn with n ≥ 3, and I ⊆ V (G) such that |I| ≥ 2. Then I is a minimum internally-locating dominating set in G if and only if the following holds: (i) u /∈ I, where u ∈ V (G) be a central vertex in G; (ii) I is a minimum internally-locating dominating set of C2n. Proof. Let G = Gn be a gear graph with n ≥ 3, vertex set V (Gn) = {u} ∪ {vi | 1 ≤ i ≤ n} ∪ {vi,i+1 | 1 ≤ i ≤ n} I. Tropico, I. Cabahug, Jr. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5918 10 of 21 and edge set E(Gn) = E(C2n) ∪ {uvi | 1 ≤ i ≤ n}, where u is the central vertex of Gn and C2n is a cycle graph of order 2n. Assume I ⊆ V (G) is a γli-set of G. Then, the central vertex u is adjacent to all vi, 1 ≤ i ≤ n but is not adjacent to any vi,i+1. Suppose u ∈ I. Then, u dominates all vi, but it does not dominate vi,i+1, which must still be dominated by other vertices in I. Removing u from I does not violate the domination or ILDS conditions. Thus, u /∈ I, as its inclusion would violate the minimality of I. Now, by definition of G, V (G) \ {u} is a C2n. Since (i) holds, and by Theorem 4, I is a γli-set of C2n. Conversely, assume that u /∈ I and I is a γli − set of C2n. Clearly, by Theorem 4, I is the γli − set of G. Corollary 5. Let G be a gear graph Gn with n ≥ 3. Then, γli(G) = n. Proof. Note that |V (Gn)| − 1 = |V (C2n)| = 2n, and also by Theorem 6, |I| = 2n, where I is the γli − set of Gn. This implies, 2n ≡ 0, 2 (mod 4). By Corollary 3, if n ≡ 0, 2 (mod 4), then γli(C2n) = n 2 . Let l ∈ Z such that l = 2n. With this, γli(Gn) = l 2 = 2n 2 = n. Hence, γli(G) = n. Theorem 7. Let G be a lollipop graph Lm,n with m ≥ 3 and n ≥ 4, and I ⊆ V (G) such that |I| ≥ 2. Then I is a minimum internally-locating dominating set in G if and only if the following holds: (i) u ∈ I and degI(u) = 0 or degI(u) = 1; (ii) for every v ∈ I, if degI(v) = 0 or 1, then degV (G)\I(v) = 1; (iii) if degI(v) = 0, then v is unique in I; and (iv) for all x ∈ V (G) \ I, x ∈ N(y) for some y ∈ I. Where, u ∈ V (Km) and incident to a bridge and v ∈ V (Pn). Proof. Let G = Lm,n be a lollipop graph with m ≥ 3 and n ≥ 4, and I ⊆ V (G) such that |I| ≥ 2. Assume I is a γli − set in G. Let u ∈ V (Km) be incident to the bridge. Since I is a γli − set, u ∈ I to dominate a vertex in Pn. If degI(u) > 1, then multiple vertices in I are unnecessarily dominated, contradicting the minimality of I. Hence, degI(u) = 0 or degI(u) = 1. Note that V (Lm,n) \ V (Km) induces a path graph Pn. Thus, by Theorem 4, (ii), (iii) and (iv) holds. I. Tropico, I. Cabahug, Jr. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5918 11 of 21 Conversely, assume conditions (i)− (iv) hold. Then, by Theorem 4, I is a γli − set in G. Corollary 6. Let G be a lollipop graph Lm,n with m ≥ 3 and n ≥ 4. Then, γli(G) =  n+2 2 if n ≡ 0, 2(mod 4) n+1 2 if n ≡ 1(mod 4) n+3 2 if n ≡ 3(mod 4) Proof. Let G be a lollipop graph Lm,n with m ≥ 3 and n ≥ 4. For convenience, V (Lm,n) = V (Km)∪V (Pn), where V (Km) = {u1, u2, . . . , um} and V (Pn) = {v1, v2, . . . , vn} such that u1v1 ∈ E(Lm,n). Now, consider the following cases: Case 1: n ≡ 0 (mod 4) Let I ⊆ V (Lm,n) with I = {u1, v2, v3, v6, v7, . . . , vn−2, vn−1}. Observe that |I| = n+2 2 . By Theorem 7, I is a γli − set. Therefore, γli(Lm,n) = |I| = n+2 2 . Case 2: n ≡ 1 (mod 4) Let I ⊆ V (Lm,n) with I = {u1, v3, v4, v7, v8, . . . , vn−2, vn−1}. Observe that |I| = n+1 2 . By Theorem 7, I is a γli − set. Therefore, γli(Lm,n) = |I| = n+1 2 . Case 3: n ≡ 2 (mod 4) Let I ⊆ V (Lm,n) with I = {u1, v1, v4, v5, . . . , vn−2, vn−1}. Observe that |I| = n+2 2 . By Theorem 7, I is a γli − set. Therefore, γli(Lm,n) = |I| = n+2 2 . Case 4: n ≡ 3 (mod 4) Let I ⊆ V (Lm,n) with I = {u1, v1, v4, v5, . . . , vn−3, vn−2, vn}. Observe that |I| = n+3 3 . By Theorem 7, I is a γli − set. Therefore, γli(Lm,n) = |I| = n+3 2 . Theorem 8. Let G be a graph with ∆(G) = 2, and I ⊆ V (T (G)) such that |I| ≥ 2. Suppose that I ⊆ V (G). Then I is an internally-locating dominating set in T (G) if and only if I is an internally-locating dominating set in V (G) such that the component of G[V (G) \ I] ̸= P2. Proof. Let G be a graph with ∆(G) = 2, and I ⊆ V (T (G)) such that |I| ≥ 2. Suppose I ⊆ V (G). Assume I is an ILDS in T (G). Then, for all u, v ∈ I, NT (G)(u) ∩ I ̸= NT (G)(u) ∩ I. Thus, NV (G)(u) ∩ I ̸= NV (G)(v) ∩ I, making I an ILS in V (G). Suppose that G[V (G) \ I] = P2, the edge corresponding to this path would not be dominated by I in T (G). This violates the domination condition in T (G), so G[V (G)\I] ̸= P2. Consequently, since I ⊆ V (G), this implies I is a dominating set in V (G). Hence, I is an ILDS in V (G). I. Tropico, I. Cabahug, Jr. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5918 12 of 21 Conversely, assume that I is an ILDS in V (G) and G[V (G) \ I] ̸= P2. Now, for all u, v ∈ I, NV (G)(u) ∩ I ̸= NV (G)(v) ∩ I. This implies NT (G)(u) ∩ I ̸= NT (G)(v) ∩ I, making I an ILS in T (G). Note that by the definition of T (G), V (T (G)) = V (G) ∪ E(G). By assumption that I is an ILDS in V (G), and since G[V (G) \ I] ̸= P2, this implies that for all ei ∈ E(G), ei ∈ N(u), for some u ∈ I. Hence, I is an ILS in T (G). Remark 1. Let G be a path graph Pn or cycle graph Cn with n = 2 or 3. Then, γli(T (G)) = γli(T (Pn)) = γli(T (Cn)) = 2. The result is straighforward and can be directly verified from the total graph of a path graph or cycle graph of order 2 or 3. Theorem 9. Let G be a path graph Pn with n ≥ 4 Then, γli(T (G)) =  4n 7 if n ≡ 0 (mod 7) ⌈ 4n 7 ⌉ if n ≡ 1, 5 (mod 7) ⌊ 4n 7 ⌋ if n ≡ 2, 3, 4, 6 (mod 7) Proof. Let G be a path graph of order n ≥ 4 and T (G) be a total graph of G. For conve- nience, let V (G) = V (Pn) = {v1, v2, . . . , vn−1, vn}, E(G) = E(Pn) = {e1, e2, e3 . . . , en−2, en−1} where e1 is the edge incident with v1 and v2, e2 is the edge incident to v2 and v3, and so on and so forth. By definition of total graph, V (T (G)) = V (G) ∪ E(G). Suppose I is an internally-locating dominating set of minimum cardinality, i.e., I is a γli − set. Now, consider the following cases: Case 1: n ≡ 0 (mod 7). Let S ⊆ V (T (G)) with S = {v2, v3, v9, v10, . . . , vn−5, vn−4} ∪ {e5, e6, e12, e13, . . . , en−9, en−8, en−2, en−1}. Observe that S is a dominating set and for all u, v ∈ S, N(u) ∩ S ̸= N(v) ∩ S. Hence, S is an ILDS. To end this, note that |S| = 4n 7 . Since I is a γli − set, |S| ≥ |I|. So, |I| ≤ |S| = 4n 7 . On the other hand, since I is a γli − set of T (G), then I must have atleast 4n 7 vertices in T (G). Hence, |I| ≥ 4n 7 . Therefore, |I| = 4n 7 . Case 2: n ≡ 1 (mod 7). Let S ⊆ V (T (G)) with S = {v2, v3, v9, v10, . . . , vn} ∪ {e5, e6, e11, e12, . . . , en−10, en−9, en−3, en−2}. I. Tropico, I. Cabahug, Jr. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5918 13 of 21 Observe that S is a dominating set and for all u, v ∈ S, N(u) ∩ S ̸= N(v) ∩ S. Hence, S is an ILDS. To end this, note that |S| = ⌈4n7 ⌉. Since I is a γli − set, |S| ≥ |I|. So, |I| ≤ |S| = ⌈4n7 ⌉. On the other hand, since I is a γli − set of T (G), then I must have atleast ⌈4n7 ⌉ vertices in T (G). Hence, |I| ≥ ⌈4n7 ⌉. Therefore, |I| = ⌈4n7 ⌉. Case 3: n ≡ 2 (mod 7). Let S ⊆ V (T (G)) with S = {v2, v3, v9, v10, . . . , vn} ∪ {e5, e6, e12, e13, . . . , en−11, en−10, en−4, en−3}. Observe that S is a dominating set and for all u, v ∈ S, N(u) ∩ S ̸= N(v) ∩ S. Hence, S is an ILDS. To end this, note that |S| = ⌊4n7 ⌋. Since I is a γli − set, |S| ≥ |I|. So, |I| ≤ |S| = ⌊4n7 ⌋. On the other hand, since I is a γli − set of T (G), then I must have atleast ⌊4n7 ⌋ vertices in T (G). Hence, |I| ≥ ⌊4n7 ⌋. Therefore, |I| = ⌊4n7 ⌋. Case 4: n ≡ 3 (mod 7). Let S ⊆ V (T (G)) with S = {v2, v3, v9, v10, . . . , vn−1} ∪ {e5, e6, e12, e13, . . . , en−12, en−11, en−5, en−4}. Observe that S is a dominating set and for all u, v ∈ S, N(u) ∩ S ̸= N(v) ∩ S. Hence, S is an ILDS. To end this, note that |S| = ⌊4n7 ⌋. Since I is a γli − set, |S| ≥ |I|. So, |I| ≤ |S| = ⌊4n7 ⌋. On the other hand, since I is a γli − set of T (G), then I must have atleast ⌊4n7 ⌋ vertices in T (G). Hence, |I| ≥ ⌊4n7 ⌋. Therefore, |I| = ⌊4n7 ⌋. Case 5: n ≡ 4 (mod 7). Let S ⊆ V (T (G)) with S = {v2, v3, v9, v10, . . . , vn−2, vn−1} ∪ {e5, e6, . . . , en−6, en−5}. Observe that S is a dominating set and for all u, v ∈ S, N(u) ∩ S ̸= N(v) ∩ S. Hence, S is an ILDS. To end this, note that |S| = ⌊4n7 ⌋. Since I is a γli − set, |S| ≥ |I|. So, |I| ≤ |S| = ⌊4n7 ⌋. On the other hand, since I is a γli − set of T (G), then I must have atleast ⌊4n7 ⌋ vertices in T (G). Hence, |I| ≥ ⌊4n7 ⌋. Therefore, |I| = ⌊4n7 ⌋. Case 6: n ≡ 5 (mod 7). Let S ⊆ V (T (G)) with S = {v2, v3, v8, v9, . . . , vn−4, vn−3} ∪ {e5, e6, . . . , en−7, en−6, en−1}. Observe that S is a dominating set and for all u, v ∈ S, N(u) ∩ S ̸= N(v) ∩ S. Hence, S is an ILDS. To end this, note that |S| = ⌈4n7 ⌉. Since I is a γli − set, |S| ≥ |I|. So, |I| ≤ |S| = ⌈4n7 ⌉. On the other hand, since I is a γli − set of T (G), then I must have atleast ⌈4n7 ⌉ vertices in T (G). Hence, |I| ≥ ⌈4n7 ⌉. Therefore, |I| = ⌈4n7 ⌉. Case 7: n ≡ 6 (mod 7). Let S ⊆ V (T (G)) with S = {v2, v3, v9, v10, . . . , vn−4, vn−3} ∪ {e5, e6, . . . , en−8, en−7, en−2, en−1}. Observe that S is a dominating set and for all u, v ∈ S, N(u) ∩ S ̸= N(v) ∩ S. Hence, S is an ILDS. To end this, note that |S| = ⌊4n7 ⌋. Since I is a γli − set, |S| ≥ |I|. So, |I| ≤ |S| = ⌊4n7 ⌋. On the other hand, since I is a γli − set of T (G), then I must have atleast ⌊4n7 ⌋ vertices in T (G). Hence, |I| ≥ ⌊4n7 ⌋. Therefore, |I| = ⌊4n7 ⌋. I. Tropico, I. Cabahug, Jr. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5918 14 of 21 Example 8. Consider Figure 7. Clearly I = {v2, v3, e5} is a dominating set. Observe that, N(v2) ∩ I = {v3}, N(v3) ∩ I = {v2}, and N(e5) ∩ I = ∅. Hence, I is an internally- locating dominating set of minimum cardinality, so γli(T (P6)) = 3. By Theorem 9, for n = 6 ≡ 6 (mod 7), γli(T (P6)) = ⌊4(n)7 ⌋ = ⌊4(6)7 ⌋ = ⌊247 ⌋ = 3. v1 v2 v3 v4 v5 v6 e1 e2 e3 e4 e5 T (P6): Figure 7: The minimum internally-locating dominating set of T (P6) Theorem 10. Let G be a cycle graph Cn with n ≥ 4 Then, γli(T (G)) =  4n 7 if n ≡ 0 (mod 7) ⌈ 4n 7 ⌉ if n ≡ 1, 3, 4, 5 (mod 7) ⌊ 4n 7 ⌋ if n ≡ 2, 6 (mod 7) Proof. LetG be a cycle graph of order n ≥ 4 and T (G) be a total graph ofG. For conve- nience, let V (G) = V (Cn) = {v1, v2, . . . , vn−1, vn}, E(G) = E(Cn) = {e1, e2, e3 . . . , en−1, en} where e1 is the edge incident with v1 and v2, e2 is the edge incident to v2 and v3, and so on, up to en which is incident with v1 and vn. By definition of total graph, V (T (G)) = V (G) ∪ E(G). Suppose I is an internally-locating dominating set of minimum cardinality, i.e., I is a γli − set. Now, consider the following cases: Case 1: n ≡ 0 (mod 7). Let S ⊆ V (T (G)) with S = {v2, v3, v9, v10, . . . , vn−5, vn−4} ∪ {e5, e6, e12, e13, . . . , en−9, en−8, en−2, en−1}. Observe that S is a dominating set and for all u, v ∈ S, N(u) ∩ S ̸= N(v) ∩ S. Hence, S is an ILDS. To end this, note that |S| = 4n 7 . Since I is a γli − set, |S| ≥ |I|. So, |I| ≤ |S| = 4n 7 . On the other hand, since I is a γli − set of T (G), then I must have atleast 4n 7 vertices in T (G). Hence, |I| ≥ 4n 7 . Therefore, |I| = 4n 7 . Case 2: n ≡ 1 (mod 7). Let S ⊆ V (T (G)) with I. Tropico, I. Cabahug, Jr. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5918 15 of 21 S = {v2, v3, v9, v10, . . . , vn} ∪ {e5, e6, e11, e12, . . . , en−10, en−9, en−3, en−2}. Observe that S is a dominating set and for all u, v ∈ S, N(u) ∩ S ̸= N(v) ∩ S. Hence, S is an ILDS. To end this, note that |S| = ⌈4n7 ⌉. Since I is a γli − set, |S| ≥ |I|. So, |I| ≤ |S| = ⌈4n7 ⌉. On the other hand, since I is a γli − set of T (G), then I must have atleast ⌈4n7 ⌉ vertices in T (G). Hence, |I| ≥ ⌈4n7 ⌉. Therefore, |I| = ⌈4n7 ⌉. Case 3: n ≡ 2 (mod 7). Let S ⊆ V (T (G)) with S = {v2, v3, v9, v10, . . . , vn} ∪ {e5, e6, e12, e13, . . . , en−11, en−10, en−4, en−3}. Observe that S is a dominating set and for all u, v ∈ S, N(u) ∩ S ̸= N(v) ∩ S. Hence, S is an ILDS. To end this, note that |S| = ⌊4n7 ⌋. Since I is a γli − set, |S| ≥ |I|. So, |I| ≤ |S| = ⌊4n7 ⌋. On the other hand, since I is a γli − set of T (G), then I must have atleast ⌊4n7 ⌋ vertices in T (G). Hence, |I| ≥ ⌊4n7 ⌋. Therefore, |I| = ⌊4n7 ⌋. Case 4: n ≡ 3 (mod 7). Let S ⊆ V (T (G)) with S = {v2, v3, v9, v10, . . . , vn−1, vn} ∪ {e5, e6, . . . , en−12, en−11, en−5, en−4}. Observe that S is a dominating set and for all u, v ∈ S, N(u) ∩ S ̸= N(v) ∩ S. Hence, S is an ILDS. To end this, note that |S| = ⌈4n7 ⌉. Since I is a γli − set, |S| ≥ |I|. So, |I| ≤ |S| = ⌈4n7 ⌉. On the other hand, since I is a γli − set of T (G), then I must have atleast ⌈4n7 ⌉ vertices in T (G). Hence, |I| ≥ ⌈4n7 ⌉. Therefore, |I| = ⌈4n7 ⌉. Case 5: n ≡ 4 (mod 7). Let S ⊆ V (T (G)) with S = {v1, v2, v8, v9, . . . , vn−3, vn−2} ∪ {e4, e5, . . . , en−7, en−6, en−1}. Observe that S is a dominating set and for all u, v ∈ S, N(u) ∩ S ̸= N(v) ∩ S. Hence, S is an ILDS. To end this, note that |S| = ⌈4n7 ⌉. Since I is a γli − set, |S| ≥ |I|. So, |I| ≤ |S| = ⌈4n7 ⌉. On the other hand, since I is a γli − set of T (G), then I must have atleast ⌈4n7 ⌉ vertices in T (G). Hence, |I| ≥ ⌈4n7 ⌉. Therefore, |I| = ⌈4n7 ⌉. Case 6: n ≡ 5 (mod 7). Let S ⊆ V (T (G)) with S = {v2, v3, v9, v10, . . . , vn−3, vn−2} ∪ {e5, e6, . . . , en−7, en−6, en−1}. Observe that S is a dominating set and for all u, v ∈ S, N(u) ∩ S ̸= N(v) ∩ S. Hence, S is an ILDS. To end this, note that |S| = ⌈4n7 ⌉. Since I is a γli − set, |S| ≥ |I|. So, |I| ≤ |S| = ⌈4n7 ⌉. On the other hand, since I is a γli − set of T (G), then I must have atleast ⌈4n7 ⌉ vertices in T (G). Hence, |I| ≥ ⌈4n7 ⌉. Therefore, |I| = ⌈4n7 ⌉. Case 7: n ≡ 6 (mod 7). Let S ⊆ V (T (G)) with S = {v2, v3, v9, v10, . . . , vn−4, vn−3} ∪ {e5, e6, . . . , en−8, en−7, en−1}. I. Tropico, I. Cabahug, Jr. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5918 16 of 21 Observe that S is a dominating set and for all u, v ∈ S, N(u) ∩ S ̸= N(v) ∩ S. Hence, S is an ILDS. To end this, note that |S| = ⌊4n7 ⌋. Since I is a γli − set, |S| ≥ |I|. So, |I| ≤ |S| = ⌊4n7 ⌋. On the other hand, since I is a γli − set of T (G), then I must have atleast ⌊4n7 ⌋ vertices in T (G). Hence, |I| ≥ ⌊4n7 ⌋. Therefore, |I| = ⌊4n7 ⌋. Example 9. Consider Figure 8. Clearly I = {v1, v2, e3} is a dominating set. Observe that, N(v1) ∩ I = {v2}, N(v2) ∩ I = {v1}, and N(e3) ∩ I = ∅. Hence, I is an internally- locating dominating set of minimum cardinality, so γli(T (C4)) = 3. By Theorem 10, for n = 4 ≡ 4 (mod 7), γli(T (C4)) = ⌈4(n)7 ⌉ = ⌈4(4)7 ⌉ = ⌈167 ⌉ = 3. v1 v2 v3 v4 e1 e2 e3 e4 T (C4): Figure 8: The minimum internally-locating dominating set in T (C4) Theorem 11. Let G be a graph with ∆(G) = 2, and I ⊆ V (T (G)) such that |I| ≥ 2. Suppose that I ⊆ V (G). Then I is an internally-locating dominating set in S(G) if and only if the following holds: (i) I is an internally-locating dominating set in V (G); and (ii) for all u ∈ I, degI(u) ̸= 0. Proof. Let G be a graph with ∆(G) = 2, and I ⊆ V (S(G)) such that |I| ≥ 2. Suppose that I ⊆ V (G). Assume I ⊆ V (G) is an ILDS in S(G). Then, by definition of S(G), I must dominate all vertices in V (S(G)) = V (G) ∪ V ′(G). In particular, I dominates V (G), as I ⊆ V (G). Also, the property of I as ILDS for S(G) holds for V (G) since I ⊆ V (G). Thus, (i) holds. I. Tropico, I. Cabahug, Jr. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5918 17 of 21 Now, if degI(u) = 0 for some u ∈ I, then u would not contribute to the domination of S(G). Specifically, u would fail to dominate u′ ∈ V ′(G), the shadow vertex corresponding to u, violating the domination property of I in S(G). Hence, degI(u) ̸= 0 for all u ∈ I. Conversely, assume I ⊆ V (G) satisfies (i) and (ii). By Condition (i), I dominates V (G). This follows, each shadow vertex v′ ∈ V ′(G) is adjacent to the neighbors of its corresponding vertex v ∈ V (G). Since I dominates V (G), it also dominates V ′(G). Hence, I dominates all of V (S(G)) = V (G) ∪ V ′(G). Now, by (i), for any distinct u, v ∈ I, their neighborhoods in V (G) satisfy N(u)∩ I ̸= N(v)∩I. This uniqueness extends to S(G) because the neighborhoods of vertices in V ′(G) are determined by their neighbors in V (G). Thus, N(u) ∩ I ̸= N(v) ∩ I for any u, v ∈ I in S(G). Finally, by (ii), degI(u) ̸= 0 for all u ∈ I, ensuring that every vertex in I actively contributes to the domination of S(G). Therefore, I satisfies the conditions to be an ILDS in S(G). Lemma 1. Let G be a path graph Pn or cycle graph Cn of order n ≥ 2 such that n ≡ 0 (mod 4). Then I ⊆ V (S(G)) with I = {v2, v3, v6, v7, . . . , vn−2, vn−1} is a minimum internally- locating dominating set in G. Proof. Let I ⊆ V (S(G)) with I = {v2, v3, v6, v7, . . . , vn−2, vn−1}. Note that I is an internally-locating dominating set in V (G) by Theorem 4. So, Theorem 11 (i) is satisfied. Additionally, for all u ∈ I, degI(u) ̸= 0. This implies, Theorem 11 (i) is satisfied. Now, removing a vertex v in I, contradicts Theorem 11 (ii). Hence, I is a γli − set. Lemma 2. Let G be a path graph Pn or cycle graph Cn of order n ≥ 2 such that n ≡ 1 (mod 4). Then I ⊆ V (S(G)) with I = {v1, v2, v5, v6, . . . , vn−4, vn−3, vn−1, vn} is a minimum internally-locating dominating set in G. Proof. Let I ⊆ V (S(G)) with I = {v1, v2, v5, v6, . . . , vn−4, vn−3, vn−1, vn}. Note that I is an internally-locating dominating set in V (G) by Theorem 4. So, Theorem 11 (i) is satisfied. Additionally, for all u ∈ I, degI(u) ̸= 0. This implies, Theorem 11 (i) is satisfied. Now, removing a vertex v in I, contradicts Theorem 11 (ii). Hence, I is a γli − set. Lemma 3. Let G be a path graph Pn or cycle graph Cn of order n ≥ 2 such that n ≡ 2 (mod 4). Then I ⊆ V (S(G)) with I = {v1, v2, v5, v6, . . . , vn−5, vn−4, vn−1, vn} is a minimum internally-locating dominating set in G. Proof. Let I ⊆ V (S(G)) with I = {v1, v2, v5, v6, . . . , vn−5, vn−4, vn−1, vn} . Note that I is an internally-locating dominating set in V (G) by Theorem 4. So, Theorem 11 (i) is satisfied. Additionally, for all u ∈ I, degI(u) ̸= 0. This implies, Theorem 11 (i) is satisfied. Now, removing a vertex v in I, contradicts Theorem 11 (ii). Hence, I is a γli − set. I. Tropico, I. Cabahug, Jr. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5918 18 of 21 Lemma 4. Let G be a path graph Pn or cycle graph Cn of order n ≥ 2 such that n ≡ 3 (mod 4). Then I ⊆ V (S(G)) with I = {v2, v3, v6, v7, . . . , vn−2, vn−1} is a minimum internally- locating dominating set in G. Proof. Let I ⊆ V (S(G)) with I = {v2, v3, v6, v7, . . . , vn−2, vn−1}. Note that I is an internally-locating dominating set in V (G) by Theorem 4. So, Theorem 11 (i) is satisfied. Additionally, for all u ∈ I, degI(u) ̸= 0. This implies, Theorem 11 (i) is satisfied. Now, removing a vertex v in I, contradicts Theorem 11 (ii). Hence, I is a γli − set. Corollary 7. Let G be a path graph Pn or cycle graph Cn with n ≥ 2 Then, γli(S(G)) = γli(S(Pn)) = γli(S(Cn)) =  n 2 if n ≡ 0 (mod 4) n+3 2 if n ≡ 1 (mod 4) n+2 2 if n ≡ 2 (mod 4) n+1 2 if n ≡ 3 (mod 4) Proof. Let G be a path graph Pn or a cycle graph Cn with n ≥ 2. For convenience, let V (G) = V (Pn) = V (Cn) = {v1, v2, v3, . . . , vn−1, vn} and V ′(G) = V ′(Pn) = V ′(Cn) = {v′1, v′2, v′3, . . . , v′n−1, v ′ n} as the set of all shadow vertices of V (G) where joining v′i to the neighbors of vi ∈ V (G). By definition of shadow graph, V (S(G)) = V (G) ∪ V ′(G). Now, observe the following cases: Case 1: n ≡ 0 (mod 4) Let I = {v2, v3, v6, v7, . . . , vn−2, vn−1}. By Lemma 1, I is a γli − set in G. Therefore, γli(G) = |I| = |{v2, v3, v6, v7, . . . , vn−2, vn−1}| = n 2 . Case 2: n ≡ 1 (mod 4) Let I = {v2,v3,v6,v7, . . . ,vn−4,vn−3,vn−1,vn} . By Lemma 2, I is a γli−set inG. Therefore, γli(G) = |I| = |{v2, v3, v6, v7, . . . , vn−4, vn−3, vn−1, vn}| = n+ 3 2 . Case 3: n ≡ 2 (mod 4) I. Tropico, I. Cabahug, Jr. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5918 19 of 21 Let I = {v2,v3,v6,v7, . . . ,vn−5,vn−4,vn−1,vn} . By Lemma 3, I is a γli−set inG. Therefore, γli(G) = |I| = |{v2, v3, v6, v7, . . . , vn−5, vn−4, vn−1, vn}| = n+ 2 2 . Case 4: n ≡ 3 (mod 4) Let I = {v2, v3, v6, v7, . . . , vn−2, vn−1}. By Lemma 4, I is a γli − set in G. Therefore, γli(G) = |I| = |{v2, v3, v6, v7, . . . , vn−2, vn−1}| = n+ 1 2 . Example 10. Consider Figure 9. Clearly I = {a, b, d, e} is a dominating set. Observe that, N(a) ∩ I = {b}, N(b) ∩ I = {a}, N(d) ∩ I = {e}, and N(d) ∩ I = {e}. Hence, I is an internally-locating dominating set of minimum cardinality, so γli(S(P5)) = 4. By Corollary 7, for n = 5 ≡ 1 (mod 4), γli(S(P5)) = n+3 2 = 5+3 2 = 8 2 = 4. a b c d e a′ b′ c′ d′ e′ S(P5): Figure 9: The minimum internally-locating dominating set in S(P5) 5. Conclusion It was shown in this paper that certain properties of this concept are identified, and characterizations of special classes of graphs are provided, including total graphs and shadow graphs with ∆(G) = 2, along with their corresponding internally-locating dom- ination numbers. The paper also examines cases where γli(G) = 2 and γ(G) = γli(G). 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