EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 1, Article Number 5648 ISSN 1307-5543 – ejpam.com Published by New York Business Global Connected Hop Roman Dominating Functions in Graphs Alkajim A. Aradais3,∗, Jerry Boy G. Cariaga1, Sergio R. Canoy, Jr.1,2 1 Department of Mathematics and Statistics, College of Science and Mathematics, MSU-Iligan Institute of Technology, 9200 Iligan City, Philippines 2 Center of Mathematical and Theoretical Physical Sciences, Premier Research Institute of Science and Mathematics, MSU-Iligan Institute of Technology, 9200 Iligan City, Philippines 3 Integrated Laboratory School, College of Education, MSU-Tawi-Tawi College of Technology and Oceanography, 7500 Bongao, Tawi-Tawi, Philippines Abstract. Let G be a connected graph. A hop Roman dominating function f : V (G) → {0, 1, 2} is a connected hop Roman dominating function (CHRDF) on G if the set {u ∈ V (G) : f(u) ̸= 0} induces a connected subgraph of G. The weight of a CHRDF f is given by ωcRh G (f) = ∑ v∈V (G) f(v) and the minimum weight among all connected hop Roman dominating functions on G, denoted γcRh(G), is the connected hop Roman domination number of G. In this paper, we show that the parameter lies between the connected hop domination number of G and twice this number. We characterize the graphs that attain small values of the parameter and determine the connected hop Roman domination number of some graphs. 2020 Mathematics Subject Classifications: 05C69 Key Words and Phrases: Connected, hop domination, hop Roman dominating function, connected hop Roman domination number 1. Introduction Motivated by the ancient Roman Empire’s military strategy, Cockayne et al. in [9] introduced and studied the parameter called Roman domination. In a sense, the concept is one of the numerous variants of the standard domination concept. Various studies have been done since the introduction of Roman domination. In particular, a significant number of variations of the parameter have already been defined and investigated (see [1], [3], [4], [7], [8], [10], [11], [12], [16], [20], [21], [24]). The concept of hop domination, one that utilizes distance two rather than unit distance, has also been widely studied since the time it was introduced in [22]. Studies in [5], [6], [13], ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5648 Email addresses: alkajimaradais@msutawi-tawi.edu.ph (A. Aradais), jerryboy.cariaga@g.msuiit.edu.ph (J. Cariaga), sergio.canoy@g.msuiit.edu.ph (S. Canoy Jr.) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) A. Aradais, J. Cariaga, S. Canoy Jr. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5648 2 of 13 [14], [15], [17], [19], [18], [23], [26], and [27] considered further the concept and some of its constructs or variants. Recently, the parameter hop Roman domination was introduced and, just like Roman domination, various modifications of the concept have also been introduced and studied (see [2], [25], and [28]). This present study considers connected hop Roman domination. Since hop domination and Roman domination have both applications in many networks (for example, to model defense strategies, communication in social networks, and management problems), this newly defined parameter can easily find its own similar applications. This among others gives added motivation for introducing and studying the said parameter. 2. Terminology and Notations Let G = (V (G), E(G)) be an undirected graph. The open neighborhood of v ∈ V (G) is the set NG(v) = {u ∈ V (G) : uv ∈ E(G)} while its closed neighborhood is the set NG[v] = {v}∪NG(v). Vertex v is an isolated vertex if NG(v) = ∅. The open neighborhood and closed neighborhood of set S ⊆ V (G) are the sets NG(S) = ∪v∈SNG(v) and NG[S] = ∪v∈SNG[u], respectively. The degree of v, denoted by degG(v), is equal to |NG(v)|. Any shortest path connecting two vertices x and y of G is called an x-y geodesic and the length of an x-y geodesic in G is the distance dG(x, y) of x and y. The diameter of G, denoted diam(G), is the maximum distance between the pair of vertices. A vertex v of G is a leaf if degG(v) = 1 and w ∈ V (G) is a support vertex if wz ∈ E(G) for some leaf z ∈ V (G). A set S ⊆ V (G) is said to be a dominating set of G if NG[S] = V (G). The minimum cardinality of a dominating set, denoted by γ(G), is called the domination number of G. A vertex v is a dominating vertex of G if NG[v] = V (G). Any dominating set of cardinality γ(G) is referred to as a γ-set of G. The open hop neighborhood of v ∈ V (G) is the set N2 G(v) = {u ∈ V (G) : dG(u, v) = 2} and its closed hop neighborhood is N2 G[v] = {v} ∪ N2 G(v). The open hop neighborhood and closed hop neighborhood of set S ⊆ V (G) are the sets N2 G(S) = ∪v∈SN 2 G(v) and N2 G[S] = {v} ∪N2 G(u), respectively. A set S ⊆ V (G) is said to be a hop dominating set of G if N2 G[S] = V (G), i.e., for each v ∈ V (G) \ S, there exists w ∈ S such that dG(v, w) = 2. A hop dominating set S is connected hop dominating if the graph ⟨S⟩ induced by S is connected. The minimum cardinality among all hop dominating (resp. connected hop dominating) sets in G is called the hop domination number (resp. connected hop domination number) of G, and is denoted by γh(G) (resp. γch(G). Any hop dominating (resp. connected hop dominating) set of cardinality γh(G) (resp. γch(G)) is called a γh-set (resp. γch-set) of G. A function f : V (G) → {0, 1, 2} is a hop Roman dominating function on G if for each u ∈ V (G) for which f(u) = 0, there exists v ∈ V (G) such that f(v) = 2 and dG(u, v) = 2. The weight of f is given by ωRh G (f) = ∑ v∈V (G) f(v). The hop Roman domination number of G, denoted by γRh(G), is the minimum weight of a hop Roman dominating function on G. A. Aradais, J. Cariaga, S. Canoy Jr. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5648 3 of 13 Let G be a connected graph. A function f : V (G) −→ {0, 1, 2} is a connected hop Roman dominating function on G provided that it satisfies the following properties: (P1) For each v ∈ V (G) with f(v) = 0, there exists w ∈ V (G) with f(w) = 2 and dG(w, v) = 2 (i.e., f is a hop Roman dominating function on G). (P2) The set {u ∈ V (G) : f(u) ̸= 0} induces a connected subgraph of G. The weight of f is given by ωcRh G (f) = ∑ v∈V (G) f(v). The minimum weight among all connected hop Roman dominating functions on G is the connected hop Roman dom- ination number γcRh(G) of G. If f is a connected hop Roman dominating function on G and ωcRh G (f) = γcRh(G), then f is called a γcRh-function on G. If f is a (con- nected) hop Roman dominating function on G, then we may write f = (V0, V1, V2) where Vj = {x ∈ V (G) : f(x) = j} for j ∈ {0, 1, 2}. Consider graph G in Figure 1. Let V0 = {a, b, c, f, g, h}, V1 = {i}, and V2 = {d, e}. Then f = (V0, V1, V2) is a γRh-function on G. On the other hand, the function g = (V ′ 0 , V ′ 1 , V ′ 2), where V ′ 0 = {a, b, c, f, g, h, i}, V ′ 1 = ∅, and V ′ 2 = {d, e, f}, is a γcRh- function on G. Therefore, γRh(G) = 5 and γcRh(G) = 6. ............... .............. .............. .............. .............. .............. ........ .................................... ............................................................................................. .................................... ................................................................................................................................. .................................... ............... .............. .............. .............. .............. .............. ........ .................................... .................................................................................................................................................... ............... .............. .............. .............. .............. .............. ........ .................................... ............................................................................................. .................................... ................................................................................................................................. .................................... ............... .............. .............. .............. .............. .............. ........ .................................... .................................................................................................................................................... ....................................a c b d e h g f i Figure 1: Graph G with γRh(G) = 5 and γcRh(G) = 6 Henceforth, the family CHRDF (G) refers to the set containing all the connected hop Roman dominating functions on G. 3. Results Proposition 1. Let G be a nontrivial connected graph and let f = (V0, V1, V2) be a γcRh- function on G. Then each of the following holds: (i) |V0| = 0 if and only if |V2| = 0. (ii) |V1| = 0 if and only if V2 is a γch-set in G. Proof. (i) The sufficiency part is clear by property (P1). Suppose |V0| = 0 and suppose |V2| ≠ ∅. Let V ′ 0 = V0, V ′ 1 = V1 ∪ V2 and V ′ 2 = ∅. Then g = (V ′ 0 , V ′ 1 , V ′ 2) ∈ CHRDF (G) and so, ωcRh G (g) = |V ′ 1 | = |V1 ∪ V2| = |V1|+ |V2| < |V1|+ 2|V2| = ωcRh G (f), A. Aradais, J. Cariaga, S. Canoy Jr. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5648 4 of 13 a contradiction. Thus, |V2| = ∅. (ii) Suppose |V1| = 0. Then V2 is a connected hop dominating set in G. Suppose that V2 is not a γch-set in G. Let V ′ 2 ⊆ V (G) be a γch-set in G. Then |V ′ 2 | < |V2|. Let V ∗ 2 = V ′ 2 , V ∗ 1 = ∅ and V ∗ 0 = V (G) \ V ∗ 2 . Then h = (V ∗ 0 , V ∗ 1 , V ∗ 2 ) ∈ CHRDF (G). Thus, ωcRh G (h) = 2|V ∗ 2 | = 2|V ′ 2 | < 2|V2| = ωcRh G (f). This is a contradiction to the assumption that f is a γcRh-function on G. Hence, V2 is a γch-set in G. Conversely, suppose that V2 is a γch-set in G. Suppose further that |V1| ≠ 0. Then γcRh(G) = |V1| + 2|V2| > 2|V2|. Let V ′′ 0 = V0 ∪ V1, V ′′ 1 = ∅ and V ′′ 2 = V2. Then V ′′ 0 ⊆ N2 G(V ′′ 2 ) and V ′′ 1 ∪ V ′′ 2 = V2 is a connected hop dominating set in G. Thus, h = (V ′′ 0 , V ′′ 1 , V ′′ 2 ) ∈ CHRDF (G) and ωcRh G (h) = 2|V ′′ 2 | = 2|V2| < |V1|+ 2|V2| = ωcRh G (f), a contradiction. Hence, |V1| = 0. Proposition 2. For any graph G of order n, it holds that γch(G) ≤ γcRh(G) ≤ min{n, 2γch(G)}. Proof. Let f = (V0, V1, V2) be a γcRh-function of G. Since V1 ∪ V2 is a connected hop dominating set of G, we have γch(G) ≤ |V1|+ |V2| ≤ |V1|+ 2|V2| = γcRh(G). Next, set V ′ 0 = V ′ 2 = ∅ and V ′ 1 = V (G). Then g1 = (V ′ 0 , V ′ 1 , V ′ 2) ∈ CHRDF (G). Thus, γcRh(G) ≤ ωcRh G (g1) = |V ′ 1 | = |V1| = n. Finally, let S be a γch-set in G and let V ′′ 1 = ∅, V ′′ 0 = V (G) \ S, and V ′′ 2 = S. Then g2 = (V ′′ 0 , V ′′ 1 , V ′′ 2 ) ∈ CHRDF (G). This implies that γcRh(G) ≤ ωcRh G (g2) = 2|V ′′ 2 | = 2γch(G). Hence, γcRh(G) ≤ min{n, 2γch(G)}. Therefore, γch(G) ≤ γcRh(G) ≤ min{n, 2γch(G)}. It is worth noting that the bounds given in Proposition 2 are sharp. Indeed, it can be verified easily that γcRh(K4) = 4 = γch(K4), γch(P3) = 2 < 3 = |V (P3)| = γcRh(P3), and γcRh(P5) = 4 = 2γch(P5) < |V (P5)|. A. Aradais, J. Cariaga, S. Canoy Jr. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5648 5 of 13 Remark 1. Let G be a connected graph. If f = (V0, V1, V2) is a γcRh-function in G, then V1 ∪ V2 need not be a γch-set in G. Remark 2. Let G be a connected graph of order n and f = (V0, V1, V2) a γcRh-function on G. If v ∈ V (G) and degG(v) = n − 1 (that is, v is a dominating vertex in G), then v ∈ V1. Proposition 3. Let G be a connected graph of order n. Then each the following statements holds: (i) γcRh(G) = 1 if and only if G = K1. (ii) γcRh(G) = 2 if and only if G = K2. (iii) γcRh(G) = 3 if and only if n ≥ 3 and G = K3 or G = K1 + H, where H is a (disconnected) graph of order n− 1 and has at least one isolated vertex. (iv) γcRh(G) = 4 if and only if n ≥ 4 and G ∈ {K4, P4, C4,K1 + P3} or one of the following conditions holds: (a) There exists w ∈ V (G) such that |NG(w)| = 2, N2 G(w) = V (G) \NG[w], and x is not a support vertex whenever x ∈ NG(w) and is a dominating vertex of G. (b) There exist adjacent vertices u, v ∈ V (G) such that |NG(u)| ≥ 2, |NG(v)| ≥ 2, N2 G(u) ∪ N2 G(v) = V (G) \ {u, v}, NG(u) ∩ NG(v) = ∅, and {u, v} is a γch-set in G. Proof. (i) Assume that γcRh(G) = 1 and f = (V0, V1, V2) is a γcRh-function on G. Then |V1| + 2|V2| = 1. This implies that V2 = ∅. By Proposition 1(i), V0 = ∅. Thus, |V (G)| = |V1| = 1. Hence, G = K1. The converse is clear. (ii) Suppose γcRh(G) = 2. Let f = (V0, V1, V2) be a γcRh-function in G. Then |V1| + 2|V2| = 2. Thus, |V2| ≤ 1. Assume that |V2| = 1. Then V1 = ∅ and, by Proposition 1(i), V0 ̸= ∅. Let V2 = {x} and let y ∈ V0. Then dG(x, y) = 2. Let z ∈ NG(x) ∩NG(y). Then z ∈ V1, a contradiction. Therefore, |V2| = 0. It follows that |V1| = 2, i.e., G = K2. Conversely, suppose G = K2 = ⟨{u, v}⟩. Let V1 = {u, v} and V0 = V2 = ∅. Then f = (V0, V1, V2) ∈ CHRDF (G). Thus, γcRh(G) ≤ ωcRh G (f) = |V1| = 2. Since G ̸= K1, γcRh(G) ≥ 2. Hence, γcRh(G) = 2. (iii) Suppose γcRh(G) = 3. Then n ≥ 3 by (i) and (ii). Let f = (V0, V1, V2) be a γch-function on G. Then |V1|+ 2|V2| = 3. Thus, |V2| ≤ 1. Consider the following cases: Case 1. |V2| = 0. Then |V0| = 0 and |V1| = |V (G)| = 3. This implies that G ∈ {K3, P3}. Case 2. |V2| = 1. Then |V0| ≠ ∅ and |V1| = 1. Let V1 = {v} and V2 = {w}. Since ⟨V1 ∪ V2⟩ is connected, uw ∈ E(G). Also, V0 = V (G)\{v, w}. Now, let u ∈ V0. By (P1), u ∈ N2 G(w). Let [u, y, w] A. Aradais, J. Cariaga, S. Canoy Jr. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5648 6 of 13 be a u− w geodesic. Since yw ∈ E(G), y /∈ V0. Hence, y ∈ V1, implying that y = v. Let H1 = ⟨V0⟩ and H = {w} ∪H1. Then G = {v} +H or, equivalently, G = K1 +H, where H is a disconnected graph with isolated vertex {w}. Conversely, suppose n ≥ 3 and G satisfies the given conditions. Then γcRh(G) ≥ 3 by (i) and (ii). Clearly, γcRh(K3) = 3. So suppose G = K1 +H and u is an isolated vertex of H. Let K1 = {v}. Then vu ∈ E(G). Since u is an isolated vertex of H, [u, v, w] is a u−w geodesic in G for every w ∈ V (G) \ {u, v}. This implies that dG(u,w) = 2 for every w ∈ V (G) \ {u, v}. Define the function f = (V0, V1, V2) where V2 = {u}, V1 = {v}, and V0 = V (G) \ {u, v}. Then f ∈ CHRDF (G) and γcRh(G) ≤ ωcRh G (f) = |V1|+ 2|V2| = 3. Therefore, γcRh(G) = 3. (iv) Suppose γcRh(G) = 4. Then n ≥ 4 by (i), (ii), and (iii). Let f = (V0, V1, V2) be a γcRh-function on G. Then |V1|+2|V2| = 4. It follows that |V2| ≤ 2. Consider the following cases: Case 1. |V2| = 0. Then V0 = ∅ and |V1| = |V (G)| = 4. Therefore, with reference to (iii), we must have G ∈ {K4, P4, C4,K1 + P3}. Case 2. |V2| = 1. Then |V1| = 2 and |V0| ≠ 0. Let V1 = {u, v} and V2 = {w}. Suppose v /∈ NG(w). Then [v, u, w] is v-w geodesic by (P2). This and (P1) would imply that [z, u, w] is a z-w geodesic for all z ∈ V (G) \ {u,w}. Let V ′ 2 = V2, V ′ 1 = {u}, V ′ 0 = V0 ∪ {v} = V (G) \ {u,w}. Then g = (V ′ 0 , V ′ 1 , V ′ 2) ∈ CHRDF (G). Thus, ωcRh G (g) = |V ′ 1 |+2|V ′ 2 | = 3, a contradiction. Thus, v ∈ NG(w). Similarly, u ∈ NG(w). Hence, u, v ∈ NG(w). This implies that degG(w) = 2, that is, |NG(w)| = 2. Moreover, N2 G(w) = V0 = V (G) \ NG[w]. Suppose z ∈ NG(w) is a dominating vertex of G. Suppose further that z is a support vertex of some leaf p. Then G = ⟨z⟩+H, where H = ⟨V (G) \ {z}⟩ is a graph with isolated vertex p. By part (iii), it follows that γcRh(G) = 3, contrary to the assumption that γcRh(G) = 4. This shows that (a) holds. Case 3. |V2| = 2. Then |V1| = 0. Let V2 = {u, v}. Then uv ∈ E(G) by (P2). Suppose |NG(u)| = 1. Then NG(u) = {v}. Since G ̸= K2, G = ⟨v⟩+H whereH = ⟨V (G)\{v}⟩ is a disconnected graph having u as an isolated vertex. By (iii), γcRh(G) = 3, a contradiction. Thus, |NG(u)| ≥ 2. Similarly, |NG(v)| ≥ 2. Suppose z ∈ NG(u) ∩NG(v). Then z ∈ V0, which is not possible. Hence, NG(u) ∩ NG(v) = ∅. Let y ∈ V (G) \ V2 = V0. By (P1), y ∈ N2 G(u) ∪ N2 G(v). Hence, N2 G(u) ∪N2 G(v) = V (G) \ {u, v}. Cleary, {u, v} is a γch-set in G. This shows that (b) holds. Conversely, suppose n ≥ 4 and suppose G ∈ {K4, P4, C4,K1 + P3. Let V (G) = {w, x, y, z} and set V1 = {w, x, y, z} and V0 = V2 = ∅. Then f = (V0, V1, V2) ∈ A. Aradais, J. Cariaga, S. Canoy Jr. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5648 7 of 13 CHRDF (G). Thus, γcRh(G) ≤ ωcRh G (f) = |V1| = 4. Since G ̸= K1 +H for any graph H having an isolated vertex, it follows from (iii) that γcRh(G) ≥ 4. Accordingly, γcRh(G) = 4. Now, suppose (a) holds, i.e., there exists w ∈ V (G) such that |NG(w)| = 2, N2 G(w) = V (G) \ NG[w], and x is not a support vertex whenever x ∈ NG(w) and is a dominating vertex of G. Let V ′ 0 = V (G)\NG[w], V ′ 1 = NG(w), and V ′ 2 = {w}. Then g = (V ′ 0 , V ′ 1 , V ′ 2) ∈ CHRDF (G) and γcRh(G) ≤ ωcRh G (g) = |V ′ 1 | + 2|V ′ 2 | = 4. The assumption that x is not a support vertex whenever x ∈ NG(w) and is a dominating vertex of G, implies that G ̸= K1 +H for any graph H having an isolated vertex. Therefore, γcRh(G) = 4. Lastly, suppose there exist adjacent vertices u, v ∈ V (G) such that |NG(u)| ≥ 2 and |NG(v)| ≥ 2, N2 G(u) ∪N2 G(v) = V (G) \ {u, v}, NG(u) ∩NG(v) = ∅ and {u, v} is a γch-set in G. Define a function h : V (G) → {0, 1, 2} by h(x) = { 2, if x ∈ {u, v}, 0, if x ∈ V (G) \ {u, v}. Then h ∈ CHRDF (G) and γcRh(G)(h) ≤ ωcRh G = h(u) + h(v) = 4. Now, suppose γ(G) = 1, say q is a dominating vertex of G. Then clearly, q /∈ V (G) \ {u, v} since N2 G(u) ∪ N2 G(v) = V (G) \ {u, v}. This implies that q ∈ {u, v}. This, however, is not possible because NG(u) ∩ NG(v) = ∅. Thus, γ(G) ̸= 1. Therefore, G ̸= K1 +H for any graph H. Therefore, γcRh(G) = 4. The following result follows from Proposition 3. Corollary 1. Let n,m be positive integers. Then (i) γcRh(Kn) = n for all n ≥ 1; (ii) γcRh(K1,n) = 3 for all n ≥ 2; (iii) γcRh(Kn,m) = 4 for all n,m ≥ 2; (iv) γcRh(P ) = 4, where P is the Petersen graph; and (v) γcRh(Fn) = 4 for all n ≥ 3. Proposition 4. For any wheel graph Wn with n ≥ 4, γcRh(Wn) = 5. Proof. Clearly, γcRh(Wn) ≥ 4. Let v0 be the hub (central) vertex of the wheel graph Wn = K1 + Cn and let V (Wn) \ {v0} = V (Cn) = {v1, v2, . . . , vn}, where Cn = [v1, v2, . . . , vn, v1]. Define a function f : V (Wn) → {0, 1, 2} by f(v) =  2, if v ∈ {v1, v2} 1, if v = v0 0, if v ∈ V (Wn) \ {v0, v1, v2}. A. Aradais, J. Cariaga, S. Canoy Jr. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5648 8 of 13 Then f ∈ CHRDF (Wn) and γcRh(Wn) ≤ ωcRh Wn (f) = f(v0) + f(v1) + f(v2) = 5. Since γch(Wn) = 3 and |NWn(x)| ≥ 3 for every x ∈ V (Wn), Wn does not satisfy (a) and (b) of Proposition 3(iv). Consequently, γcRh(Wn) = 5. Proposition 5. For any path Pn of order n ≥ 1, γcRh(Pn) =  4, if n = 5, 6 6, if n = 7 n, if n ̸= 5, 6, 7. Proof. For n = 1, 2, 3, the result follows from Propositon 3 (i), (ii), (iii). By Propo- sition 3 (iv), γcRh(Pn) = 4 for n = 4, 5, 6. Let n ≥ 7 and let Pn = [x1, x2, ..., xn]. Let f = (V0, V1, V2) be a γcRh-function on Pn. Since V1 ∪ V2 is a connected hop dom- inating set, V0 ⊆ {x1, x2, xn−1, xn}. Suppose first that n = 7. Since the function g = ({x1, x2, x6, x7},∅, {x3, x4, x5}) is a connected hop dominating function on P7, and P7 does not satisfy any of the conditions given in Proposition 3, it follows that 5 ≤ γcRh(P7) ≤ 6. Now, suppose that γcRh(P5) = |V1|+2|V2| = 5. If V0 = ∅, then |V2| = 0 and |V1| = 7. This implies that γcRh(P7) = 7 which is not possivle. Thus, |V0| ≠ 0 and 1 ≤ |V2| ≤ 2. Suppose |V2| = 1. Then |V0| = 3 and |V1| = 3. If x1 ∈ V1 ∪ V2, then V1 ∪ V2 = {x1, x2, x3, x4} since ⟨V1 ∪ V2⟩ is connected. Hence, x7 ∈ V0 and dP7(x7, y) ̸= 2 for all y ∈ V2, a contra- diction. Thus, x1 ∈ V0. Similarly, x7 ∈ V0. It follows that V1 ∪ V2 is {x2, x3, x4, x5} or {x3, x4, x5, x6}. This implies that x1 /∈ N2 P7 (V2) or x7 /∈ N2 P7 (V2), a contradiction. This forces |V2| = 2 and |V1| = 1. It is routine to show that this also leads to a contradiction. Therefore, γcRh(P7) = 6. Next, suppose that n ≥ 8. If |V0| = 0, then |V2| = 0 and |V1| = n. Hence, γcRh(Pn) = n. Suppose |V0| = 1. Then V0 = {x1} or V0 = {xn}. Assume that V0 = {x1}. Then V2 = {x3} and V1 = V (Pn) \ {x1, x3}. This yields γcRh(Pn) = ωcRh Pn (f) = |V1|+ 2|V2 = (n− 2) + 2 = n. If |V0| = 2, then V0 = {x1, x2} or V0 = {x1, xn} or V0 = {xn−1, xn}. It follows that V2 = {x3, x4} or V2 = {x3, xn−2} or V2 = {xn−3, xn−2}, respectively. If |V0| = 3, then V0 = {x1, x2, xn} or V0 = {x1, xn−1, xn}. Hence, V2 = {x3, x4, xn−2} or V2 = {x3, xn−3, xn−2}, respectively. Finally, if V0 = {x1, x2, xn−1, xn}, then V2 = {x3, x4, xn−3, xn−2}. It can easily be shown that any of these cases will imply that γcRh(Pn) = ωcRh Pn (f) = |V1|+ 2|V2 = (n− 2) + 2 = n. This proves the assertion. A. Aradais, J. Cariaga, S. Canoy Jr. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5648 9 of 13 Proposition 6. For any cycle Cn of order n ≥ 3, γcRh(Cn) =  4, if n = 4, 5, 6 6, if n = 7 n, if n ̸= 5, 6, 7. Proof. By Proposition 3(iii), γcRh(C3) = 3 and by Proposition 3(iv), γcRh(Cn) = 4 for n = 4, 5, 6. Next, let n ≥ 7 and let Cn = [v1, v2, ..., vn, v1]. Let f = (V0, V1, V2) be a γcRh- function on Cn. Since V1 ∪ V2 is a connected hop dominating set, ⟨V0⟩ is connected and |V0| ≤ 4. Suppose first that n = 7. Since the function g = ({v1, v2, v3, v4},∅, {v5, x6, v7}) is a connected hop dominating function on C7, γcRh(C7) ≤ 6. If |V0| = 0, then |V2| = 0 and |V1| = 7. If |V0| = 1, then |V2| = 1 and |V1| = 5. If |V0| = 2, then |V2| = 2 and |V1| = 3 and if |V0| = 3, then |V2| = 3 and |V1| = 1. Any of these four cases will yield γcRh(C7) = 7 which is not possible. Therefore, |V0| = 4. Hence, |V2| = 3 and |V1| = 0. It follows that γcRh(C7) = 6. Finally, suppose that n ≥ 8. Clearly, if |V0| = j, then |V2| = j and |V1| = n− 2j. Therefore, γcRh(Cn) = n− 2j + 2j = n. Proposition 7. Let G = Kn1,n2,...,nk be the complete k-partite graph with 2 ≤ n1 ≤ n2 ≤ · · · ≤ nk, where k ≥ 2. Then γcRh(G) = 2k. Proof. Let Sn1 , Sn2 , . . . , Snk be the partite sets in G. Pick vnj ∈ Snj for each j ∈ [k], where [k] = {1, 2, . . . , k}, and let S = {vnj : j ∈ [k]}. Set V0 = V (G) \ S, V1 = ∅, and V2 = S. Then f = (V0, V1, V2) ∈ CHRDF (G) and γcRh(G) ≤ 2|V2| = 2k. Now let g = (W0,W1,W2) be a γcRh-function on G. Let M = {j ∈ [k] : nj ≥ 3}. Suppose |W2| = 0. Then |W0| = 0 and |W1| = V (G). It follows that γcRh(G) = |V (G)| =∑k j=1 nj . The value is 2k ifM = ∅ and strictly greater than 2k ifM ̸= ∅. Hence, |W2| ≠ 0 if M ̸= ∅. So suppose |W2| ̸= 0. Let R = {j ∈ [k] : W2 ∩ Snj ̸= ∅}. Since g is a Roman dominating function on G, it follows that W0 = ∪j∈R[Snj \ [(W2 ∩Snj )∪ (W1 ∩Snj )] and W1 = [∪j∈[k]\RSnj ] ∪ [∪j∈R(Snj ∩W1)]. It follows that γcRh(G) = ∑ j∈[k]\R Snj + ∑ j∈R (Snj ∩W1) + 2 ∑ j∈R (Snj ∩W2) ≥ 2(k − |R|) + 2|R| = 2k. This establishes the desired equality. Lemma 1. Let G be a connected graph of order n. Then γch(G) = n if and only if G = Kn. Proof. Suppose γch(G) = n and suppose for a contradiction that G ̸= Kn. Then diam(G) ≥ 2. Pick any two vertices x and y of G such that dG(x, y) = diam(G). Then x and y are non-cut vertices of G. Let [x1, x2, . . . , xk], where x = x1 and y = xk, be an x-y geodesic in G. Then k ≥ 3 and dG(x1, x3) = 2. Put S = V (G) \ {x}. Since x is a non-cut vertex, it follows that ⟨S⟩ is connected. Therefore, S is a connected hop dominating set in G and γch(G) ≤ |S| = n− 1, contrary to our assumption. Thus, G = Kn. A. Aradais, J. Cariaga, S. Canoy Jr. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5648 10 of 13 For the converse, suppose G = Kn and let S be a gammach-set in G. Since S is a hop dominating set, V (Kn) ⊆ S. Therefore, S = V (G) and γch(G) = n. Proposition 8. Let G be any graph of order n. Then γch(G) = γcRh(G) if and only if G = Kn. Proof. Suppose γch(G) = γcRh(G) and let f = (V0, V1, V2) be a γcRh-function in G. Since γch(G) ≤ |V1| + |V2| ≤ |V1| + 2|V2| = γcRh(G) and γch(G) = γcRh(G), |V1| + |V2| = |V1| + 2|V2|. This implies that V2 = ∅ and thus, V0 = ∅. Hence, γcRh(G) = |V1| = n = γch(G). By Lemma 1, G = Kn. For the converse, suppose that G = Kn. By Lemma 1, γch(G) = n. By Proposition 2, γcRh(G) = n. This shows that γch(G) = γcRh(G). The next result is immediate from Proposition 3. Proposition 9. Let G and H be connected graphs. Then γcRh(G +H) ≥ 2 and each of the following holds: (i) γcRh(G+H) = 2 if and only if G and H are trivial graphs. (ii) γcRh(G+H) = 3 if and only if G = K1 and H = K2 (or H = K1 and G = K2) or G = K1 and H is a graph with at least one isolated vertex (or H = K1 and G is a graph with at least one isolated vertex). (iii) If G and H are non-trivial graphs and each contains an isolated vertex, then γcRh(G+H) = 4. Proposition 10. Let G and H be connected graphs. Then 2 ≤ γcRh(G+H) ≤ γcRh(G) + γcRh(H). Proof. Since G + H is nontrivial, γcRh(G + H) ≥ 2. Let f = (V0, V1, V2) and g = (V ′ 0 , V ′ 1 , V ′ 2) be γcRh-functions onG andH, respectively. Define a function h = (V ′′ 0 , V ′′ 1 , V ′′ 2 ) for which V ′′ 0 = V0 ∪V ′ 0 , V ′′ 1 = V1 ∪V ′ 1 and V ′′ 2 = V2 ∪V ′ 2 . Then h ∈ CHRDF (G+H) and γcRh(G+H) ≤ ωcRh G+H = |V ′′ 1 |+ 2|V ′′ 2 | = |V1 ∪ V ′ 1 |+ 2|V2 ∪ V ′ 2 | = |V1|+ |V ′ 1 |+ 2|V2|+ 2|V ′ 2 | = |V1|+ 2|V2|+ |V ′ 1 |+ 2|V ′ 2 | = γcRh(G) + γcRh(H). (1) Remark 3. The upper bound given in Proposition 10 is tight. However, strict inequality is attainable. To see this, let G = P3 = [a, b, c] and H = P2 = [p, q]. Then γcRh(G) = 3 by Proposition 3(iii) and γcRh(H) = 2 by Proposition 3(ii). Let f = (V0, V1, V2) be a γcRh-function on G + H. Since b, p, and q are dominating vertices of G + H, it follows A. Aradais, J. Cariaga, S. Canoy Jr. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5648 11 of 13 from Remark 1 that b, p, q ∈ V1. If a, c ∈ V1 ∪ V2, then V1 = {a, b, c, p, q}. Hence, γcRh(G + H) = 5. Suppose one of a and c is in V0, say V0 = {a}. Then necessarily, c ∈ V2. It follows that V1 = {b, p, q} and γcRh(G + H) = |V1| + 2|V2| = 5. Therefore, γcRh(G+H) = γcRh(G) + γcRh(H) = 5. Clearly, the upper bound is also attained when G = Km and H = Kn. Next, to show that strict inequality is also possible, consider G = P4 = [a, b, c, d] and H = P4 = [x, y, z, w]. Then γcRh(G) = γcRh(H) = 4 by Proposition 3(iv). Now set V0 = {a, b, x, y}, V1 = {c, z}, and V2 = {d,w}. Then f = (V0, V1, V2) ∈ CHRDF (G+H). It is easy to see that f is a γcRh-function on G +H. Therefore, γcRh(G +H) = 6 < 8 = γcRh(G) + γcRh(H). Proposition 11. There are infinitely many graphs G and H such that γcRh(G+H) = 5. In particular, if G = Pn, where n ≥ 4, and H is any non-trivial graph with at least one trivial component, then γcRh(G+H) = 5. Proof. Let n ≥ 4 and let G = Pn = [x1, x2, . . . , xn]. Let H1, H2, . . . ,Hk be the com- ponents of H and suppose that H1 = ⟨p⟩. Let V0 = (V (Pn) \ {xn−1, xn}) ∪ (∪k j=2V (Hj)), V1 = {xn−1} and V2 = {p, xn}. Then g = (V0, V1, V2) is a connected hop Roman dominat- ing function on G+H. It follows that γcRh(G+H) ≤ ωcRh G (f) = 5. Since G+H does not satisfy any of the conditions given in Proposition 3, it follows that γcRh(G+H) = 5. 4. Conclusion Connected hop Roman domination has been defined and investigated for some graphs. There are still a lot of aspects that can be explored and studied for this parameter. For any two graphs G and H, this initial study has only provided sharp lower and upper bounds for the join G + H. It was, however, shown that these bounds may not be attained. It is conjectured that the exact value of the parameter for the join G+H can be described by defining yet another parameter. Furthermore, connected hop Roman domination can also be investigated for other graphs under binary operations and for the complexity of the connected hop Roman dominating function problem. Acknowledgements The authors would like to thank the referees for the their comments and suggestions which led to the improvement of the paper. Also, the authors would like to thank the Department of Science and Technology - Accelerated Science and Technology Human Re- source Development Program (DOST-ASTHRDP)-Philippines, and MSU-Iligan Institute of Technology (Iligan City, Philippines) for funding this research. A. Aradais, J. Cariaga, S. Canoy Jr. / Eur. J. Pure Appl. 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