EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 15, No. 2, 2022, 635-645 ISSN 1307-5543 – ejpam.com Published by New York Business Global Monophonic Eccentric Domination Numbers of Graphs Sergio R. Canoy, Jr.1, Anabel E. Gamorez2,∗ 1 Department of Mathematics and Statistics, College of Science and Mathematics, Center for Graph Theory, Algebra and Analysis-PRISM, MSU-Iligan Institute of Technology, Iligan City 9200, Philippines 2 Department of Mathematics and Statistics, College of Science and Mathematics, Western Mindanao University, Baliwasan, Zamboanga City 7000, Philippines Abstract. Let G be a (simple) undirected graph with vertex and edge sets V (G) and E(G), respectively. A set S ⊆ V (G) is a monophonic eccentric dominating set if every vertex in V (G) \S has a monophonic eccentric vertex in S. The minimum size of a monophonic eccentric dominating set in G is called the monophonic eccentric domination number of G. It shown that the absolute difference of the domination number and monophonic eccentric domination number of a graph can be made arbitrarily large. We characterize the monophonic eccentric dominating sets in graphs resulting from the join, corona, and lexicographic product of two graphs and determine bounds on their monophonic eccentric domination numbers. 2020 Mathematics Subject Classifications: 05C69 Key Words and Phrases: Monophonic, eccentric, domination, join, corona, lexicographic product 1. Introduction In a recent study, Santhakumaran and Titus in [3] and [4] defined monophonic distance and obtained some results related to the concept. Using monophonic paths and monophonic distance-related concepts, Titus et al. in [6], and [7] defined and studied a variation of domination called monophonic eccentric domination and the correspond- ing monophonic eccentric domination number. Titus and Fancy [5] also introduced total monophonic eccentric dominating set. The authors mentioned that the monophonic ec- centric domination number and total monophonic eccentric domination number find useful applications in channel assignment problems in radio technologies and in molecular prob- lems in theoretical chemistry. Recently, Gamorez and Canoy in [1] and [2] also made use of the monophonic distance and related concepts to construct a topology on a vertex set of a given undirected graph. Some characterizations were obtained and subbasic open sets on graphs resulting from some binary operations were determined. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v15i2.4354 Email addresses: sergio.canoy@g.msuiit.edu.ph (S. Canoy, Jr.) anabel.gamorez@wmsu.edu.ph (A. Gamorez) https://www.ejpam.com 635 © 2022 EJPAM All rights reserved. S. Canoy, Jr., A. Gamorez / Eur. J. Pure Appl. Math, 15 (2) (2022), 635-645 636 2. Terminology and Notations For any two vertices u and v in an undirected connected graph G, the distance dG(u, v) is the length of a shortest path joining u and v. The open neighborhood of a point u is the set NG(u) consisting of all points v which are adjacent to u. The closed neighborhood of u is NG[u] = NG(u) ∪ {u}. A chord of a path P in a graph G is an edge joining two non-adjacent vertices of P . A path P in a graph G is called a monophonic path if it is chordless. For any two vertices u and v in a connected graph G, the monophonic dis- tance dmG (u, v) from u to v is defined as the length of a longest u-v monophonic path in G. The monophonic eccentricity emG (v) of a vertex v in G is the maximum monophonic distance from v to a vertex of G. The monophonic radius radm(G) of graph G is given by radm(G) = min{emG (v) : v ∈ V (G)} and the monophonic diameter diamm(G) of G is given by diamm(G) = max{emG (v) : v ∈ V (G)}. A vertex w in G is a monophonic eccentric vertex of a vertex v in G if emG (v) = dmG (w, v). In this case, we say that w is a monophonic eccentric neighbor of v. The set consisting of all the monophonic eccentric vertices of v ∈ V (G) will be denoted by Nm G (v), i.e., Nm G (v) = {w ∈ V (G) : emG (v) = dmG (w, v)}. Here, Nm G [v] = Nm G (v) ∪ {v}. A set S ⊆ V (G) is a monophonic eccentric dominating set (total monophonic eccentric dominating set) of G if each w ∈ V (G) \ S (resp. w ∈ V (G)) has a monophonic eccentric vertex in S. The smallest size of a monophonic eccentric dominating (total monophonic eccentric dominating) set of G, denoted by γme(G) (resp. γtme(G)), is called the monophonic eccentric domination number (resp. total monophonic eccen- tric domination number) of G. Any monophonic eccentric dominating (total monophonic eccentric dominating) set of G of size γme(G) (resp. γtme(G)) is called a minimum mono- phonic eccentric dominating set or a γme-set (resp. minimum total monophonic eccentric dominating set or γtme-set) of G. Let G be a connected graph with diamm(G) ≥ 3. A set S ⊆ V (G) is a d3m-monophonic eccentric set of G if for each u ∈ V (G) \ S with emG (u) ≥ 3, there exists w ∈ S such that emG (u) = dmG (w, u). The minimum cardinality of a d3m-monophonic eccentric set of G, denoted by µ3 me(G), is called the d3m-monophonic eccentric number of G. The join of two graphs G and H, denoted by G + H is the graph with vertex set V (G+H) = V (G)∪ V (H) and edge set E(G+H) = E(G)∪E(H)∪ {uv : u ∈ V (G), v ∈ V (H)}. The corona of graphs G and H, denoted by G ◦H, is the graph obtained fromG by taking a copy Hv of H and forming the join 〈v〉 + Hv = v + Hv for each v ∈ V (G). The lexicographic product of graphs G and H, denoted by G[H], is the graph with vertex set V (G[H]) = V (G) × V (H) and (v, a)(u, b) ∈ E(G[H]) if and only if either uv ∈ E(G) or u = v and ab ∈ E(H). Note that any non-empty set C ⊆ V (G)× V (H) can be written as C = ⋃ x∈S [{x} × Tx], where S ⊆ V (G) and Tx ⊆ V (H) for each x ∈ S. 3. Results Theorem 1. Let G be a connected graph of order n ≥ 1. Then γme(G) = 1 if and only if G = Kn or there exists v ∈ V (G) satisfying the following properties: S. Canoy, Jr., A. Gamorez / Eur. J. Pure Appl. Math, 15 (2) (2022), 635-645 637 (i) V (G) \NG[v] = {w ∈ V (G) : dmG (v, w) = 2} 6= ∅; (ii) NG(v) ⊆ NG(w) for each w ∈ V (G) \NG[v]; and (iii) dmG (x, y) ≤ 2 for all x, y ∈ V (G) \NG[v]. Proof. Suppose γme(G) = 1 and let S = {v} be a monophonic eccentric dominating set of G. If G = Kn, then we are done. So suppose that G 6= Kn. Suppose NG[v] = V (G). Since G 6= Kn, there exist vertices a, b ∈ V (G) such that dG(a, b) = 2 ≤ dmG (a, b). It follows that v is not a monophonic eccentric vertex of a, contrary to our assumption of S. Thus, V (G)\NG[v] 6= ∅. Now let w ∈ V (G)\NG[v]. Since v is a monophonic eccentric vertex of w, we have em(w) = dmG (v, w) ≥ 2. Suppose dmG (v, w) 6= 2, say [v1, v2, ..., vk], where v1 = v, vk = w and k ≥ 4, is a v-w monophonic path. Since dG(v2, w) ≥ 2, this would imply that v is not a monophonic eccentric vertex of v2, a contradiction. Thus, dmG (v, w) = 2. This shows that (i) holds. Next, let z ∈ NG(v) and let w ∈ V (G) \ NG[v]. Since v is a monophonic eccentric vertex of z, it follows that dG(z, w) = 1, that is, z ∈ NG(w). This shows that (ii) holds. Finally, let x, y ∈ V (G) \ NG[v]. If dmG (x, y) ≥ 3, then v is not a monophonic eccentric vertex of x, a contradiction. Thus, dmG (x, y) ≤ 2, showing that (iii) holds. For the converse, suppose first that G = Kn. Then, clearly, γme(G) = 1. Next, suppose that there exists v ∈ V (G) satisfying conditions (i), (ii), and (iii). Let S0 = {v}. By (ii), v is a monophonic eccentric vertex of every element of NG(v). Let w ∈ V (G) \ NG[v]. Then dmG (v, w) = 2 by (i). Further, by (iii), it follows that v is a monophonic eccentric vertex of w. Therefore, γme(G) = |S0| = 1. Theorem 2. Let G1, G2, ..., Gk be the distinct components of G with k ≥ 2 and let H = K1 +G = 〈v〉+G. (i) If diamm(Gi) ≤ 2 for each i ∈ {1, 2, ..., k} and one of the components is trivial, then γme(H) = 1. (ii) If diamm(Gi) ≤ 2 for each i ∈ {1, 2, ..., k} and none of the components is trivial, then γme(H) = 2. Proof. (i) Let Gj be a trivial component of G. Set S = V (Gj) = {w}. Clearly, NH(w) = {v}. Since diamm(Gi) ≤ 2 for each i ∈ {1, 2, ..., k}, the conditions given in Theorem 1 are satisfied. Therefore, γme(H) = 1. (ii) Since none of the components is trivial, none of the vertices of H satisfies the conditions in Theorem 1. It follows that γme(H) ≥ 2. Pick wi ∈ V (Gi) for i = 1, 2 and let S = {w1, w2}. Clearly, w1 is a monophonic eccentric vertex of v. Let z ∈ V (G) \ S. Suppose z /∈ V (G1) ∪ V (G2). Since diamm(Gi) ≤ 2 for each i ∈ {1, 2, ..., k}, it follows that w1 and w2 are monophonic eccentric vertices of z in H. Suppose z ∈ V (G1). By assumption, dmGi (z, w1) ≤ 2. Hence, emH(z) = dmH(z, w2) = 2, that is, w2 is a monophonic eccentric vertex of z in H. Similarly, w1 is a monophonic eccentric vertex of z in H if z ∈ V (G2). This shows that S is a monophonic eccentric dominating set of H. Therefore, γme(H) = |S| = 2. S. Canoy, Jr., A. Gamorez / Eur. J. Pure Appl. Math, 15 (2) (2022), 635-645 638 Theorem 3. Let G be a connected non-complete graph and let K1 = 〈v〉. Then S is a monophonic eccentric dominating set of K1 +G if and only if S ∩ V (G) is a monophonic eccentric dominating set of G. Proof. Suppose S is a monophonic eccentric dominating set of K1 + G. Since G is non-complete, SG = S ∩ V (G) 6= ∅. Let w ∈ V (G) \ SG. If dG(w, x) = 1 for every x ∈ S, then every element of SG is a monophonic eccentric vertex of w. Suppose dG(w, y) 6= 1 for some y ∈ S. Then emK1+G(w) = emG (w) ≥ dmG (w, y) ≥ 2. Since S is a monophonic eccentric dominating set of K1 + G, there exists a monophonic eccentric vertex z ∈ S of w. Since dmK1+G(w, v) = 1, z 6= v. Thus, z ∈ SG and emG (w) = dmG (z, w). Hence, S ∩ V (G) is a monophonic eccentric dominating set of G. For the converse, suppose that SG = S ∩ V (G) is a monophonic eccentric dominating set of G. Let u ∈ V (K1 + G) \ S. If u = v, then every element of S is a monophonic eccentric vertex of u in K1 + G. Suppose u 6= v. Since SG is a monophonic eccentric dominating set of G and u ∈ V (G) \ SG, there exists p ∈ SG such that emG (u) = dmG (p, u). Hence, emK1+G(u) = dmK1+G(p, u). This proves that S is a monophonic eccentric dominating set of K1 +G. The next result is a consequence of Theorem 3 and the fact that γme(H) = 1 for every complete graph H. Corollary 1. Let G be a connected graph. Then γme(K1 +G) = γme(G). Theorem 4. Let G1, G2, ..., Gk be the distinct components of G with k ≥ 2 and let H = K1 + G = 〈v〉 + G. Suppose RG = {j ∈ {1, 2, ..., k} : diamm(Gj) ≥ 3} 6= ∅. Then S is a monophonic eccentric dominating set of H if and only if Sj = S ∩ V (Gj) is a d3m-monophonic eccentric set of Gj for each j ∈ RG and, in addition, S ∩ V (Gt) 6= ∅ for some t ∈ {1, 2, ..., k} \ RG whenever |RG| = 1 and there exists p ∈ V (Gr) \ Sr such that emGr (p) = 1 or emGr (p) = 2 and dmGr (p, w) = 1 for all w ∈ Sr, where RG = {r}. Proof. Suppose S is a monophonic eccentric dominating set of H and let j ∈ RG. Let u ∈ V (Gj) \ Sj with emGj (u) ≥ 3. Then by assumption, there exists w ∈ S such that emH(u) = dmH(w, u). Since emH(u) = emGj (u) ≥ 3, it follows that w ∈ Sj and that emGj (v) = dmGj (w, u). This shows that Sj is a a d3m-monophonic eccentric set of Gj for each j ∈ RG. Suppose now that |RG| = 1, say RG = {r}. Suppose there exists p ∈ V (Gr) \ Sr such that emGr (p) = 1 or emGr (p) = 2 and dmGr (p, w) = 1 for all w ∈ Sr, where RG = {r}. Since emH(p) = 2 and S is a monophonic eccentric dominating set of H, there exist t ∈ {1, 2, ..., k} \RG and x ∈ S ∩ V (Gt) such that emH(p) = dmH(p, x) = 2. This shows that S ∩ V (Gt) 6= ∅ for some t ∈ {1, 2, ..., k} \RG. For the converse, suppose that Sj = S ∩ V (Gj) is a d3m-monophonic eccentric set of Gj for each j ∈ RG and, in addition, S ∩ V (Gt) 6= ∅ for some t ∈ {1, 2, ..., k} \ RG whenever |RG| = 1 and there exists p ∈ V (Gr)\Sr such that emGr (p) = 1 or emGr (p) = 2 and dmGr (p, w) = 1 for all w ∈ Sr, where RG = {r}. Let z ∈ V (H)\S. If v /∈ S and z = v, then every element of S is a monophonic eccentric vertex of z in H. Suppose that z ∈ V (G). S. Canoy, Jr., A. Gamorez / Eur. J. Pure Appl. Math, 15 (2) (2022), 635-645 639 If |RG| ≥ 2, then by assumption, there exists q ∈ S (q ∈ Si or q ∈ Sj , where i, j ∈ RG and i 6= j) such that emH(z) = dmH(z, q). Suppose |RG| = 1, say RG = {r}. Assume first that z ∈ V (Gr). If emGr (z) ≥ 3, then emGr (z) = emH(z) and so z has a monophonic eccentric vertex in H (in Gr) by assumption. Suppose emGr (z) = 2. If dmGr (p, w) = 2 for some w ∈ Sr, then w is a monophonic eccentric vertex of z in H. Suppose dmGr (p, w) = 1 for all w ∈ Sr. By assumption, S∩V (Gt) 6= ∅ for some t ∈ {1, 2, ..., k}\RG. Then every element of S∩V (Gt) is a monophonic eccentric vertex of z in H. If emGr (z) = 1, then every element of S∩V (Gt) is a monophonic eccentric vertex of z in H because emH(z) = 2. Next, suppose that z ∈ Gi for i 6= r. Then emH(z) = 2. Hence, every element of Sr is a monophonic eccentric vertex of z in H. Therefore, S is a monophonic eccentric dominating set of H. Corollary 2. Let G1, G2, ..., Gk be the distinct components of G with k ≥ 2 and let H = K1 +G = 〈v〉+G. Suppose RG = {j ∈ {1, 2, ..., k} : diamm(Gj) ≥ 3} 6= ∅. (i) If |RG| ≥ 2, then γme(H) = ∑ j∈RG µ3 m(Gj). (ii) If RG = {r} and γme(H) 6= µ3 m(Gr), then γme(H) = µ3 m(Gr) + 1. Proof. (i) Suppose |RG| ≥ 2. Let Sj be a d3m-monophonic eccentric set of Gj such that µ3 m(Gj) = |Sj | for each j ∈ RG and set S = ∪j∈RG Sj . Then S is a monophonic eccentric dominating set of H by Theorem 4. Hence, γme(H) ≤ |S| = ∑ j∈RG µ3 m(Gj). Next, suppose that S0 is a minimum monophonic eccentric dominating set of H. Let S′ j = S0∩V (Gj) for each j ∈ RG. By Theorem 4, S′ j is a d3m-monophonic eccentric set of Gj for each j ∈ RG. Since |S′ j | ≥ µ3 m(Gj) for each j ∈ RG, γme(H) = |S0| ≥ ∑ j∈RG µ3 m(Gj). This proves the assertion. (ii) Suppose RG = {r} and γme(H) 6= µ3 m(Gr). Let S be a minimum monophonic eccentric dominating set of H. Then S ∩ V (Gr) is a d3m-monophonic eccentric set of Gr by Theorem 4. If S ∩ V (Gr) = S, then µ3 m(Gr) < |S| = γme(H) by assumption. If S ∩ V (Gr) 6= S, again, µ3 m(Gr) ≤ |S ∩ V (Gr)| < |S| = γme(H) by assumption. Thus, µ3 m(Gr) + 1 ≤ γme(H). Next, let SG be a minimum d3m-monophonic eccentric set of Gr. Let t ∈ {1, 2, ..., k} \ {r} and choose any q ∈ V (Gt). Let S0 = SG ∪ {q}. Then S0 is a monophonic eccentric dominating set of H by Theorem 4. This implies that γme(H) ≤ |S0| = µ3 m(Gr) + 1. Therefore, γme(H) = µ3 m(Gr) + 1. Theorem 5. Let G and H be connected non-complete graphs. Then S is a monophonic eccentric dominating set of G + H if and only if S = SG ∪ SH , where SG and SH are monophonic eccentric dominating sets of G and H, respectively. Proof. Suppose S is a monophonic eccentric dominating set of G + H. Let SG = S ∩ V (G) and SH = S ∩ V (H). Since G and H are non-complete graphs, SG 6= ∅ and SH 6= ∅. Let v ∈ V (G) \ SG. If dmG (v, x) = 1 for all x ∈ V (G) \ {v}, then every element of SG is a monophonic eccentic vertex of v in G. Suppose dmG (v, y) 6= 1 for some y ∈ V (G) \ {v}. Since S is a monophonic eccentric dominating set of G+H, there exists q ∈ S such that emG+H(v) = dmG+H(v, q). Since dmG+H(v, h) = 1 for all h ∈ SH , it S. Canoy, Jr., A. Gamorez / Eur. J. Pure Appl. Math, 15 (2) (2022), 635-645 640 follows that q ∈ SG and emG (v) = dmG (v, q). This implies that SG is a monophonic eccentric dominating set of G. Similarly, SH is a monophonic eccentric dominating set of H. For the converse, suppose that S = SG ∪ SH , where SG and SH are monophonic eccentric dominating sets of G and H, respectively. Let x ∈ V (G + H) \ S. Suppose x ∈ V (G). Then x ∈ V (G) \ SG. By assumption, there exists w ∈ SG such that emG (x) = dmG (x,w). It follows that emG+H(x) = dmG+H(x,w). Similarly, if x ∈ V (H), then there exists u ∈ SH ⊆ S such that emG+H(x) = dmG+H(x, u). Therefore, S is a monophonic eccentric dominating set of G+H. Corollary 3. Let G and H be connected non-complete graphs. Then γme(G+H) = γme(G) + γme(H). The next result shows that the absolute difference of the domination number the monophonic eccentric domination number can be made arbitrarily large. Theorem 6. Let n be a positive integer. Then the following statements hold: (i) There exists a connected graph G1 such that γ(G1)− γme(G1) = n. (ii) There exists a connected graph G2 such that γme(G2)− γ(G2) = n. Proof. (i) Consider the corona G1 = Pn+2 ◦ K1 of Pn+2 = [x1, x2, ..., xn+2] and K1 in Figure 1. Clearly, S1 = {x1, x2, ..., xn+1, xn+2} is a minimum dominating set and S2 = {a, b} is a minimum monophonic eccentric dominating set of G1. Thus, γ(G1)− γme(G1) = n. .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... ............................................................................ ............................................................................ ............................................................................ ............................................................................ ............................................................................ ............................................................................................... ............................ ........ ........ ........ ........ ........ ........ ........ ........ ... ......... ........ ........ ........ ........ ........ ........ ........ ........ ... ......... ........ ........ ........ ........ ........ ........ ........ ........ ... ......... ........ ........ ........ ........ ........ ........ ........ ........ ... ......... ........ ........ ........ ........ ........ ........ ........ ........ ... ......... ........ ........ ........ ........ ........ ........ ........ ........ ... ......... ........ ........ ........ ........ ........ ........ ........ ........ ... ......... ........ ........ ........ ........ ........ ........ ........ ........ .... . . • • x1 x2 x3 x4 x5 x6 xn+1 xn+2 b a Figure 1: A graph G1 with γ(G1) = n+ 2 and γme(G1) = 2 (ii) Consider the graph G2 = K1+(∪n+2 j=1Hj), where Hj = P4 for each j ∈ {1, . . . , n+1}. Clearly, γ(G2) = 1. Now RG2 = {1, 2, ..., n + 1} (see Theorem 4) and µ3 m(Hj) = µ3 m(P4) = 1 for each j ∈ RG2 . Hence, γme(G2) = ∑ j∈RG µ3 m(Hj) = n + 1 by Corollary 2. Thus, γme(G2)− γ(G2) = n. This proves the assertion. Theorem 7. Let G and H be any connected non-trivial graphs. Then S is a monophonic eccentric dominating set of G ◦H if and only if S = A ∪ (∪v∈V (G)Sv), where A ⊆ V (G) and Sv ⊆ V (Hv) for each v ∈ V (G), and satisfies the following conditions: (i) If v ∈ V (G) \A, then Sw 6= ∅ for some w ∈ V (G) with emG (v) = dmG (v, w). S. Canoy, Jr., A. Gamorez / Eur. J. Pure Appl. Math, 15 (2) (2022), 635-645 641 (ii) If x ∈ V (H) \ Sv and emHv(x) < emG (v) + 2, then Sw 6= ∅ for some w ∈ V (G) with emG (v) = dmG (v, w). (iii) If x ∈ V (H) \ Sv and emHv(x) = emG (v) + 2, then there exists y ∈ Sv such that emHv(x) = dmHv(x, y) or Sw 6= ∅ for some w ∈ V (G) with emG (v) = dmG (v, w). (iv) If x ∈ V (H) \ Sv and emHv(x) > emG (v) + 2, then there exists y ∈ Sv such that emHv(x) = dmHv(x, y). Proof. Suppose S is a monophonic eccentric dominating set of G ◦ H. Let A = S∩V (G) and Sv = S∩V (Hv) for each v ∈ V (G). Let v ∈ V (G). If v ∈ V (G)\A, then emG◦H(v) = emG (v)+1. Hence, by assumption, there exist w ∈ V (G) with emG (v) = dmG (v, w) and q ∈ Sw such that emG◦H(v) = dmG◦H(v, q). This shows that (i) holds. Again, since S is a monophonic eccentric dominating set of G ◦H, it is routine to show that (ii), (iii) and (iv) hold. For the converse, suppose that S is the given set and satisfies the given conditions. Let x ∈ V (G ◦ H) \ S and let v ∈ V (G) such that x ∈ v + Hv. If x = v, then Sw 6= ∅ for some w ∈ V (G) with emG (v) = dmG (v, w) by (i). It follows that every element of Sw is a monophonic eccentric vertex of x in G◦H. Suppose x ∈ V (Hv)\Sv. If emHv(x) > emG (v)+2, then Sv contains a monophonic eccentric vertex of x in G◦H by (iv). If emHv(x) ≤ emG (v)+2, then x has a monophonic eccentric vertex in G ◦ H by (ii) and (iii). Therefore, S is a monophonic eccentric dominating set of G ◦H. Theorem 8. Let G and H be any connected non-trivial graphs such that radm(H) > diamm(G) + 2. Then S is a monophonic eccentric dominating set of G ◦H if and only if Sv = S ∩ V (Hv) is a monophonic eccentric dominating set of Hv for each v ∈ V (G). Moreover, γme(G ◦H) = |V (G)|γme(H). Proof. Let v ∈ V (G) and let Sv = S ∩ V (Hv). Let x ∈ V (Hv) \ Sv. Since radm(H) > diamm(G) + 2, emHv(x) > emG (v) + 2. By Theorem 7(iv), there exists y ∈ Sv such that emHv(x) = dmHv(x, y) = dmG◦H(x, y). This shows that Sv is a monophonic eccentric dominating set of Hv. For the converse, suppose that Sv is a monophonic eccentric dominating set of Hv for each v ∈ V (G). Let z ∈ V (G ◦H) \ S and let w ∈ V (G) such that z ∈ w + V (Hw). Since radm(H) > diamm(G) + 2, the conditions in Theorem 7 are satisfied by S. Therefore, S is a monophonic eccentric dominating set of G ◦H. Next, let Dv be a minimum monophonic eccentric dominating set of Hv for each v ∈ V (G). Then S0 = ∪v∈V (G)Dv is a minimum monophonic eccentric dominating set of G ◦H. Thus, γme(G ◦H) = |S0| = |V (G)|γme(H). For vertex v ∈ V (G), we denote by Nm G (v) the set of all monophonic eccentric vertices of v, i.e., Nm G (v) = {w ∈ V (G) : emG (v) = dmG (v, w)}. Let G be a connected graph. Denote by Vm(G) a smallest set of vertices of G satisfying the properties: S. Canoy, Jr., A. Gamorez / Eur. J. Pure Appl. Math, 15 (2) (2022), 635-645 642 (A) For each v ∈ Vm(G) there exists w ∈ V (G) such that v ∈ Nm G (w), and (B) |Vm(G) ∩Nm G (u)| = 1 for each u ∈ V (G). As an example, consider the graph G obtained from C4 = [a, b, c, d, a] by adding the pendant edge ae. The set {c, e} is the smallest subset of G satisfying properties (A) and (B). Thus, Vm(G) = {c, e}. Theorem 9. Let G and H be any connected non-trivial graphs such that diamm(H) < radm(G) + 2. Then S is a monophonic eccentric dominating set of G ◦H if and only if Sv 6= ∅ for each v ∈ Vm(G) having Sw 6= V (Hw) for some w ∈ V (G) with v ∈ Nm G (w), where Su = S∩V (Hu) for each u ∈ V (G). Moreover, γme(G◦H) = |Vm(G)|. Proof. Suppose S is a monophonic eccentric dominating set of G ◦ H. Let v ∈ Vm(G) and Sv = S∩V (Hv). Then Qv = {y ∈ V (G) : v ∈ Nm G (y)} 6= ∅ by property (A) of Vm(G). Suppose that Sw 6= V (Hw) for some w ∈ Qv, say z ∈ V (Hw)\Sw. By property (B) of Vm(G), it follows that |Vm(G)∩Nm G (w)| = {v}. From the assumption that diamm(H) < radm(G)+2, it follows that emHw(z) < emG (w)+2. Hence, emG◦H(z) = emG (w)+2 = dmG (w, v)+2. Since S is a monophonic eccentric dominating set of G ◦H, Theorem 7(ii) guarantees the existence of q ∈ Sv such that emG◦H(z) = dmG◦H(z, q), showing that Sv 6= ∅. For the converse, suppose that the given condition holds. Let z ∈ V (G ◦H) \ S and let w ∈ V (G) such that z ∈ V (w +Hw). Let v ∈ Nm G (w) ∩ Vm(G). Since diamm(H) < radm(G) + 2 and Sv 6= ∅ by assumption, every element of Sv is a monophonic eccentric vertex of z. Since z was arbitrarily chosen, it follows that S is a monophonic eccentric dominating set of G ◦H. Next, choose any point xv ∈ V (Hv) for each v ∈ Vm(G) and let S0 = {xv : v ∈ Vm(G)}. Then S0 is a minimum monophonic eccentric dominating set of G ◦H. Thus, γme(G ◦H) = |S0| = |Vm(G)|. Theorem 10. Let G and H be non-trivial connected graphs such that radm(G) > diamm(H). Then C = ⋃ x∈S [{x}×Tx], where S ⊆ V (G) and Tx ⊆ V (H) for each x ∈ S, is a monophonic eccentric dominating set of G[H] if and only if the following hold: (i) S is a monophonic eccentric dominating set of G. (ii) For each x ∈ S such that Tx 6= V (H), S ∩Nm G (x) 6= ∅. Proof. Suppose C is a monophonic eccentric dominating set of G[H] and let v ∈ V (G) \ S. Pick any a ∈ V (H). Then (v, a) /∈ C and so by assumption of C, there exists (w, b) ∈ C such that emG[H]((v, a)) = dmG[H]((v, a), (w, b)). It follows that v 6= w and emG (v) = dmG (v, w), i.e., w ∈ S ∩Nm G (v). This shows that S is a monophonic eccentric dominating set of G. Next, let x ∈ S with Tx 6= V (H). Let p ∈ V (H) \ Tx. Then (x, p) ∈ V (G[H]) \ C. Hence, there exists (z, q) ∈ Nm G[H]((x, p))∩C. Since radm(G) > diamm(H), it follows that S. Canoy, Jr., A. Gamorez / Eur. J. Pure Appl. Math, 15 (2) (2022), 635-645 643 dmG (x, z) > dmH(p, q). Hence, dmG[H]((z, q), (x, p)) = dmG (x, z) and z ∈ S ∩ Nm G (x), showing that (ii) holds. For the converse, suppose that (i) and (ii) hold. Let (v, a) ∈ V (G[H]) \ C. If v /∈ S, then there exists w ∈ Nm G (v) ∩ S by (i). Let d ∈ Tw. Then (w, d) ∈ C. Now, because radm(G) > diamm(H), it follows that emG[H]((v, a)) = dmG[H]((v, a), (w, d)) = dmG (w, v) = emG (v). Suppose v ∈ S. Then a /∈ Tv, i.e., Tv 6= V (H). By (ii), it follows that there exists z ∈ S ∩Nm G (v). Pick any b ∈ Tz. Then (z, b) ∈ C and emG[H]((v, a)) = dmG[H]((v, a), (z, b)) = dmG (z, v) = emG (v). Therefore, C is a monophonic eccentric dominating set of G[H]. Corollary 4. Let G and H be non-trivial connected graphs such that radm(G) > diamm(H). Then γme(G[H]) = γtme(G). Proof. Let S be a γtme-set of G and let p ∈ V (H). For each x ∈ S, let Tx = {p}. Then C = ⋃ x∈S [{x} × Tx] = S × {p} is a monophonic eccentric dominating set of G[H] by Theorem 10. Thus, γme(G[H]) ≤ |C| = |S| = γtme(G). Next, let C0 = ⋃ x∈S0 [{x}×Rx] be a γme-set of G[H]. Then S0 is a monophonic eccentric dominating set of G by Theorem 10(i). If S0 is a total monophonic eccentric dominating set, then γme(G[H]) = |C0| ≥ |S0| ≥ γtme(G). Suppose S0 is not a total monophonic eccentric dominating set. Then there exists y ∈ S0 such that Nm G (y) ∩ S0 = ∅. By Theorem 10(ii), Ry = V (H). Let S1 = {v ∈ S0 : Nm G (v) ∩ S0 = ∅}. Again, Rv = V (H) for each v ∈ S1 by Theorem 10(ii). For each v ∈ S1, choose a vertex zv ∈ Nm G (v) and set S2 = {zv : v ∈ S1}. Then S2 ∩ S0 = ∅ and |S1| ≥ |S2|. Clearly, S∗ = S0 ∪ S2 is a total monophonic eccentric dominating set of G and we have S. Canoy, Jr., A. Gamorez / Eur. J. Pure Appl. Math, 15 (2) (2022), 635-645 644 γme(G[H]) = |C0| = ∑ x∈S0 |Rx| = ∑ x∈S0\S1 |Rx|+ ∑ x∈S1 |Rx| = ∑ x∈S0\S1 |Rx|+ |V (H)||S1| ≥ ∑ x∈S0\S1 |Rx|+ 2|S1| ≥ ∑ x∈S0\S1 |Rx|+ (|S1|+ |S2|) ≥ |S0 \ S1|+ S1 + S2 = |S∗| ≥ γtme(G). Accordingly, γme(G[H]) = γtme(G). Theorem 11. Let G and H be non-trivial connected graphs such that radm(H) > diamm(G). Then C = ⋃ x∈S [{x} × Tx], where S ⊆ V (G) and Tx ⊆ V (H) for each x ∈ S, is a monophonic eccentric dominating set of G[H] if and only if (i) S = V (G) and (ii) Tx is a monophonic eccentric dominating set of H for each x ∈ V (G). Proof. Suppose C is a monophonic eccentric dominating set of G[H]. Suppose S 6= V (G), say x ∈ V (G)\S. Pick any a ∈ V (H). Then (v, a) /∈ C. As C is a monophonic eccentric dominating set of G[H], there exists (w, b) ∈ C such that emG[H]((v, a)) = dmG[H]((v, a), (w, b)). However, the assumption that radm(H) > diamm(G) implies that emG[H]((v, a)) = emH(a) = dmH(a, b) > emG (v). This is impossible because w 6= x. Thus, S = V (G), showing that (i) holds. Let x ∈ V (G). If Tx = V (G), then it is a monophonic eccentric dominating set of H. Suppose Tx 6= V (H) and let q ∈ V (H) \ Tx. Since (x, q) ∈ V (G[H]) \ C and emG[H]((x, q)) = emH(q), it follows that there exists p ∈ Tx ∩ Nm H (q). Hence, Tx is a monophonic eccentric dominating set of H. This shows that (ii) holds. For the converse, suppose that (i) and (ii) hold. Let (z, a) ∈ V (G[H]) \ C. Since S = V (G), it follows that a /∈ Tx. As Tx is a monophonic eccentric dominating set of H according to (ii), there exists b ∈ Tx such that emH(a) = dmH(a, b). With the assumption that radm(H) > diamm(G), it follows that emG[H]((z, a)) = emH(a) = dmH(a, b) = dmG[H]((z, a), (z, b)), where (z, b) ∈ C. Therefore, C is monophonic eccentric dominating set of G[H]. REFERENCES 645 The next result is an immediate consequence of Theorem 11. Corollary 5. Let G and H be non-trivial connected graphs such that radm(H) > diamm(G). Then γme(G[H]) = |V (G)|γme(H). Conclusion: Monophonic paths and monophonic distance-related concepts had been used to define monophonic eccentric dominating set and monophonic eccentric domina- tion number of a graph. It was shown that the absolute difference of the domination number and the monophonic eccentric domination number can be made arbitrarily large. Monophonic eccentric dominating sets in the join, corona, and lexicographic product of two graphs were characterized and, under some conditions, their monophonic eccentric domination numbers were subsequently determined. Several aspects of the concept (e.g. its complexity) and the corresponding parameter remains to be investigated. Moreover, other variants of the concept may as well be introduced and studied. Acknowledgements The authors would like to thank the referees for reviewing the paper and for the comments and suggestions they provided. Also, the authors would like to extend their gratefulness to the MSU-Iligan Institute of Technology, Philippines and Western Mindanao State University, Philippines for the steadfast support these institutions have afforded them. References [1] A. Gamorez and S. Canoy Jr. On a topological space generated by monophonic eccen- tric neighborhoods of a graph. Eur. J. Pure Appl. Math., 14(3):695–705, 2021. [2] A. Gamorez and S. Canoy Jr. Subbasic open sets in graphs generated by monophonic eccentric neighborhoods. Journal of Discrete Mathematical Sciences and Cryptography, 24(7):1989–2000, 2021. [3] A. Santhakumaran and P. Titus. Monophonic distance in graphs. 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