EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 1, 2023, 261-270 ISSN 1307-5543 – ejpam.com Published by New York Business Global Distance k-Cost Effective Sets in the Corona and Lexicographic Product of Graphs Julius G. Caadan1,∗, Rolando N. Paluga2, Imelda S. Aniversario3 1 Surigao del Norte State University, 8400 Surigao City, Philippines 2 Department of Mathematics, College of Mathematics and Natural Sciences, Caraga State University , 8600, Ampayon, Butuan City City, Philippines 3 Department of Mathematics and Statistics, College of Science and Mathematics, Mindanao State University-Iligan Institute of Technology, 9200 Iligan City, Philippines Abstract. Let G be a connected graph and k ≥ 1 be an integer. The open k-neighborhood Nk G(v) of v ∈ V (G) is the set Nk G(v) = {u ∈ V (G) \ {v} : dG(u, v) ≤ k}. A set S of vertices of G is called distance k-cost effective of G if for every vertex u in S, |Nk G(u) ∩ (V (G) \ S)| − |Nk G(u) ∩ S| ≥ 0. The maximum cardinality of a distance k-cost effective set of G is called the upper distance k-cost effective number of G. In this paper, we characterized the distance k-cost effective sets in the corona and lexicographic product of two graphs. Consequently, the bounds or the exact values of the upper distance k-cost effective numbers of these graphs are obtained. 2020 Mathematics Subject Classifications: 05C76, 05C12 Key Words and Phrases: Distance k-cost effective set, upper distance k-cost effective number, distance k-domonating set, very distance k-cost effective set, corona, lexicographic product 1. Introduction Let G be a connected simple graph with vertex and edge sets V (G) and E(G), respec- tively. The basic concepts of graph here are adapted from [2]. Let v ∈ V (G). The open neighborhood NG(v) of v in G is the set NG(v) = {u ∈ V (G) : uv ∈ E(G)}. The degree degG(v) of a vertex v ∈ V (G) is the cardinality of NG(v). The minimum degree of G is δ(G) = min{degG(v) : v ∈ V (G)} and the maximum degree of G is ∆(G) = max{degG(v) : v ∈ V (G)}. The distancce dG(u, v) between vertices u and v in G is the length of the shortest path from vertex u to vertex v in G. The diameter diam(G) of G is the maximum distance between any two vertices in G. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v16i1.4381 Email addresses: juliusgcaadan@gmail.com (Julius G. Caadan), rnpaluga@carsu.edu.ph (Rolando N. Paluga), imelda.aniversario@g.msuiit.edu.ph (Imelda S. Aniversario) https://www.ejpam.com 261 © 2023 EJPAM All rights reserved. Julius G. Caadan, Rolando N. Paluga, Imelda S. Aniversario / Eur. J. Pure Appl. Math, 16 (1) (2023), 261-270 262 Let k ≥ 1 be positive integer and v ∈ V (G). The open k-neighborhood Nk G(v) of vertex v is the set of all vertices u of G such that 0 < dG(u, v) ≤ k. That is, Nk G(v) = {u ∈ V (G) : 0 < dG(u, v) ≤ k}. The distance k-degree of v in G , denoted by degkG(v), is the cardinality of Nk G(v). The minimum distance k-degree of G, denoted by δk(G), is given by δk(G) = min{degkG(v) : v ∈ V (G)} and the maximum distance k-degree of G, denoted by ∆k(G), is given by ∆k(G) = max{degkG(v); v ∈ V (G)}. Note that deg1G(v) =degG(v), δ1(G) = δ(G), and ∆1(G) = ∆(G). Let G be a connected graph. Haynes et al. in [7] defined a vertex v ∈ S ⊆ V (G) as cost effective if |NG(v) ∩ (V (G) \ S)| − |NG(v) ∩ S| ≥ 0. A set S ⊆ V (G) is called cost effective if every vertex v ∈ S is a cost effective. Paluga et al. in [3] applied the distance k version for this concept. Accordingly, a nonempty set S ⊆ V (G) is a distance k-cost effective if for every v ∈ V (G), |Nk G(v) ∩ (V (G) \ S)| − |Nk G(v) ∩ S| ≥ 0. The maximum cardinality of a distance k-cost effective set in G is called upper distance k- cost effective number of G and is denoted by αk ce(G). A distance k-cost effective set in G of cardinality αk ce(G) is called an upper distance k-cost effective set and is simply called αk ce- set in G. For example, for any integer n ≥ 3 and if k = 2, S is a distance 2-cost effective set in Pn if |S| ≤ ⌊2n3 ⌋. Thus, α2 ce(Pn) = ⌊2n3 ⌋. The concept of cost effective set in graph was introduced by Haynes et al. in [7]. In 2018, Chellali et al. in [4] established a generalization of this concept. However, Paluga et al. [3] considered distance concept for the cost effective set. For some investigations of the cost effective concept, we refer the readers to see [6, 9, 11]. For some practical application of distance concept, we refer the readers to [1, 4, 5, 10, 12]. In this paper, we characterized the distance k-cost effective sets in the corona and lexicographic product of two graphs. As direct consequences, we determined the bounds or the exact values of the upper distance k-cost effective numbers of these graphs. 2. Results 2.1. Preliminary Results In this section, we present a characterization of a distance k-cost effective set in G. Some examples of the upper distance k-cost effective number of simple graphs are given. Moereover, we obtain a relationship between upper distance k-cost effective set and dis- tance k-dominating set in G. Theorem 1. Let G be a connected simple graph and k ≥ diam(G). Then S is a distance k-cost effective set in G if and only if |S| ≤ ⌊ |V (G)|+1 2 ⌋. Proof: Let G be a connected simple graph and k ≥ diam(G). Suppose S is a distance k-cost effective set in G. Then for each u ∈ S, |Nk G(u) ∩ (V (G) \ S)| − |Nk G(u) ∩ S| = | ( V (G) \ {u} ) ∩ (V (G) \ S)| − | ( V (G) \ {u} ) ∩ S| = |V (G)|+ 1− 2|S| ≥ 0. Julius G. Caadan, Rolando N. Paluga, Imelda S. Aniversario / Eur. J. Pure Appl. Math, 16 (1) (2023), 261-270 263 Thus, |S| ≤ ⌊ |V (G)|+1 2 ⌋. Conversely, suppose that |S| ≤ ⌊ |V (G)|+1 2 ⌋. Then |S| ≤ |V (G)|+1 2 . Now, |Nk G(u) ∩ (V (G) \ S)| − |Nk G(u) ∩ S| = |V (G)|+ 1− 2|S| ≥ |V (G)|+ 1− [ |V (G)|+ 1 ] = 0. Thus, S is a distance k-cost effective set in G. Corollary 1. Let G be a connected graph and k ≥ diam(G). Then αk ce(G) = ⌊ |V (G)|+1 2 ⌋. Corollary 2. Let G be a connected graph and k ≥ 2 be an integer. Then i. αk ce(Kn) = ⌊n+1 2 ⌋, for positive integer n. ii. αk ce(Km.n) = ⌊m+n+1 2 ⌋, for positive integers m and n. iii. αk ce(Fn) = ⌊n+2 2 ⌋, for integer n ≥ 3. iv. αk ce(Wn) = ⌊n+2 2 ⌋, for integer n ≥ 4. Let G be a connected graph and k ≥ 1 be an integer. Henning et al. in [8] defined distance k-dominating set of G. Accordingly, a set S ⊆ V (G) is said to be a distance k- dominating set of G if for every v ∈ V (G) \S, there exists u ∈ S such that dG(u, v) ≤ k . Theorem 2. Every upper distance k-cost effective set in a connected graph G is a distance k-dominating set in G. Proof: Suppose S is an upper distance k-cost effective set in G but not a distance k-dominating set in G. Then there exists u ∈ V (G) \ S such that dG(u, s) > k, for all s ∈ S. Let A = S ∪ {u} and x ∈ A. Suppose x ̸= u, i.e., x ∈ S. Note that u /∈ Nk G(x). Then |Nk G(x) ∩ (V (G) \ A)| − |Nk G(x) ∩ A| = |Nk G(x) ∩ (V (G) \ S)| − |Nk G(x) ∩ S| ≥ 0. Suppose x = u. Then |Nk G(x)∩(V (G)\A)|−|Nk G(x)∩A| ≥ 0. Thus, A is a distance k-cost effective set in G. This is a contradiction since S is an upper distance k-cost effective set in G. Therefore, every upper distance k-cost effective set in a connected graph G is a distance k-dominating set in G. 2.2. Corona of Graphs This section provides a neccesary condition for a distance k-cost effective set in the corona of two graphs. Correspondingly, a lower bound for the upper distance k-cost effec- tive number of the corona of graphs is determined. The corona G ◦H of two graphs G and H is the graph obtained by taking one copy of G of order n and n copies of H, and then joining every vertex of the ith copy of H Julius G. Caadan, Rolando N. Paluga, Imelda S. Aniversario / Eur. J. Pure Appl. Math, 16 (1) (2023), 261-270 264 to the ith vertex of G. For every v ∈ V (G), denote Hv the copy of H whose vertices are attached one by one to the vertex v. Subsequently, denote by v +Hv the subgraph of the corona G ◦H corresponding to the join ⟨{v}⟩+Hv, v ∈ V (G). Theorem 3. Let G and H be connected graphs and k ≥ 1 be an integer. (i) If k = 1, 2 and Sx is a distance k-cost effective set in Hx, for every x ∈ V (G), then S = ⋃ x∈V (G) Sx is a distance k-cost effective set in G ◦H. (ii) If k ≥ 3 and |Sx| ≤ |V (H)| ( δk−2(G)+1 ) +δk−1(G)+2 2 ( ∆k−2(G)+1 ) , for every x ∈ V (G), then S =⋃ x∈V (G) Sx is a distance k-cost effective set in G ◦H. Proof: Suppose Sx is a distance k-cost effective set in Hx, for every x ∈ V (G). Let S =⋃ x∈V (G) Sx and u ∈ S. Then there exists a ∈ V (G) such that u ∈ Sa. Since Sa is a distance k-cost effective set in Ha, for all a ∈ V (G), |Nk Ha (u)∩ (V (Ha) \ Sa)| − |Nk Ha (u)∩ Sa| ≥ 0. if k = 1, we have |N1 G◦H(u) ∩ ( V (G ◦H) \ S ) | = |N1 Ha (u) ( V (Ha) \ Sa ) |+ 1 and |N1 G◦H(u) ∩ S| = |N1 Ha (u) ∩ Sa| Thus, |N1 G◦H(u) ∩ ( V (G ◦ H) \ S ) | − |N1 G◦H(u) ∩ S| ≥ 0. Hence, S is a distance 1-cost effective set in G ◦H. If k = 2, then |N2 G◦H(u) ∩ (V (G ◦H) \ S)| = |N2 Ha (u) ∩ ( V (Ha) \ Sa ) |+ degG(a) + 1 and |N2 G◦H(u) ∩ S| = |N2 Ha (u) ∩ Sa|. Thus, |N2 G◦H(u) ∩ (V (G ◦ H) \ S)| − |N2 G◦H(u) ∩ S| ≥ 0. Hence, S is a distance 2-cost effective set in G ◦H. (ii) Let k ≥ 3 be an integer and u ∈ S. Then there exists a ∈ V (G) such that u ∈ Sa. Now, |Nk G◦H(u) ∩ ( V (G ◦H) \ S ) | − |Nk G◦H(u) ∩ S| = [ degk−1 G (a) + 1 ] + |V (H) \ Sa| + ∑ x∈Nk−2 G (a) ∣∣∣(V (H) \ Sx )∣∣∣− ∑ x∈Nk−2 G (a) |Sx| − (|Sa| − 1) = [ degk−1 G (a) + 1 ] + |V (H)| − |Sa|+ ∑ x∈Nk−2 G (a) ( |V (H)| − |Sx| ) − ∑ x∈Nk−2 G (a) |Sx| − |Sa|+ 1 Julius G. Caadan, Rolando N. Paluga, Imelda S. Aniversario / Eur. J. Pure Appl. Math, 16 (1) (2023), 261-270 265 = degk−1 G (a) + 2 + |V (H)| − 2|Sa|+ ∑ x∈Nk−2 G (a) ( |V (H)| − 2|Sx| ) ≥ degk−1 G (a) + 2 + |V (H)| − 2|Sp|+ |Nk−2 G (a)||V (H)| − 2|Nk−2 G (a)||Sp|,where |Sp| = max{|Sx| : x ∈ V (G)} = degk−1 G (a) + 2 + |V (H)|+ degk−2 G (a)|V (H)| − 2|Sp| − 2degk−2 G (a)||Sp| = degk−1 G (a) + 2 + |V (H)| ( degk−2 G (a) + 1 ) − 2 ( degk−2 G (a) + 1 ) |Sp| ≥ degk−1 G (a) + 2 + |V (H)| ( degk−2 G (a) + 1 ) −2 ( degk−2 G (a) + 1 )[ |V (H)| ( δk−2(G) + 1 ) + δk−1(G) + 2 2 ( ∆k−2(G) + 1 ) ] ≥ degk−1 G (a) + 2 + +|V (H)| ( degk−2 G (a) + 1 ) − ( degk−2 G (a) + 1 )[ |V (H)| ( degk−2 G (a) + 1 ) + degk−1 G (a) + 2( degk−2 G (a) + 1 ) ] = degk−1 G (a) + 2 + +|V (H)| ( degk−2 G (a) + 1 ) − [ |V (H)| ( degk−2 G (a) + 1 ) + degk−1 G (a) + 2 ] = 0. Thus, S is a distance k-cost effective set in G ◦H. Corollary 3. Let G and H be connected graphs and k ≥ 1 be an integer. Then (i) αk ce(G ◦H) ≥ |V (G)|αk ce(H), for k = 1, 2. (ii) αk ce(G ◦H) ≥ |V (G)| |V (H)| ( δk−2(G)+1 ) +δk−1(G)+2 2 ( ∆k−2(G)+1 ) , for k ≥ 3. 2.3. Lexicographic Product of Graphs This section provides a neccesary condition for a distance k-cost effective set in the lexicographic product of two graphs. Consequently, a lower bound for the upper distance k-cost effective number of this graph is given. The lexicographic productG[H] of two graphsG andH is the graph with V (G[H]) = V (G)× V (H) and (u, v)(u′, v′) ∈ E(G[H]) if and only if either uu′ ∈ E(G) or u = u′ and vv′ ∈ E(H). Let (u, v) ∈ S and k ≥ 1 be an integer. Then |Nk G[H](u, v)∩ (V (G[H]) \ S)| = |Nk Hu (v)∩ (V (H) \ Tu)|+ |Nk G(u)∩ (V (G) \A)||V (H)| + ∑ x∈Nk G(u)∩A |V (H) \ Tx| (1) Julius G. Caadan, Rolando N. Paluga, Imelda S. Aniversario / Eur. J. Pure Appl. Math, 16 (1) (2023), 261-270 266 and |Nk G[H](u, v) ∩ S| = ∑ x∈Nk G(u)∩A |Tx|+ |Nk Hu (v) ∩ Tu|. (2) Theorem 4. Let G and H be connected graphs and k ≥ 2 be an integer. Let A be an αk ce-set in G. Let S = ⋃ a∈A({a} × Ta) such that |{a} × Ta| ≤ |V (H)|+1 2 , for each a ∈ A. Then S is a distance k-cost effective set in G[H]. Proof: Let k ≥ 2 be an integer and A be an αk ce-set in G. Let S = ⋃ a∈A({a} × Ta) such that |{a} × Ta| ≤ |V (H)|+1 2 , for each a ∈ A. Let (u, v) ∈ S. Then using equations (1) and (2), we have |Nk G[H](u, v) ∩ S| = ∑ x∈Nk G(u)∩A |Tx|+ |Nk Hu (v) ∩ Tu| = ∑ x∈Nk G(u)∩A |Tx|+ |Tu| − 1. and |Nk G[H](u, v) ∩ (V (G[H]) \ S)| =|Nk Hu (v) ∩ (V (H) \ Tu)|+ |Nk G(u) ∩ (V (G) \A)||V (H)| + ∑ x∈Nk G(u)∩A |V (H) \ Tx| = |V (H)| − |Tu|+ |Nk G(u) ∩ (V (G) \A)||V (H)| + ∑ x∈Nk G(u)∩A [ |V (H)| − |Tx| ] . Hence, |Nk G[H](u, v) ∩ (V (G[H]) \ S)| − |Nk G[H](u, v) ∩ S| = |Nk G(u) ∩ (V (G) \A)||V (H)| + ∑ x∈Nk G(u)∩A [ |V (H)| − 2|Tx| ] + |V (H)| − 2|Tu|+ 1 ≥ |Nk G(u) ∩ (V (G) \A)||V (H)|+ ∑ x∈Nk G(u)∩A [ |V (H)| − 2|V (H)| ] + |V (H)| − 2|Tu|+ 1 = |Nk G(u) ∩ (V (G) \A)||V (H)| − |Nk G(u) ∩A)||V (H)| + |V (H)| − 2|Tu|+ 1 = [ |Nk G(u) ∩ (V (G) \A)| − |Nk G(u) ∩A| ] |V (H)|+ |V (H)| − 2|Tu|+ 1 ≥ [ |Nk G(u) ∩ (V (G) \A)| − |Nk G(u) ∩A| ] |V (H)|+ |V (H)| Julius G. Caadan, Rolando N. Paluga, Imelda S. Aniversario / Eur. J. Pure Appl. Math, 16 (1) (2023), 261-270 267 − [ |V (H)|+ 1 ] + 1 = [ |Nk G(u) ∩ (V (G) \A)| − |Nk G(u) ∩A| ] |V (H)|. Since A is an αk ce-set in G, [ |Nk G(u) ∩ (V (G) \ A)| − |Nk G(u) ∩ A| ] |V (H)| ≥ 0. Thus, |Nk G[H](u, v) ∩ (V (G[H]) \ S)| − |Nk G[H](u, v) ∩ S| ≥ 0. Therefore, S is a distance k-cost effective set in G[H]. Corollary 4. Let G and H be connected graphs and k ≥ 2 be an integer. Then αk ce(G[H]) ≥ |V (H)|+1 2 αk ce(G) . Theorem 5. Let G and H be connected graphs. Let A be an α1 ce-set in G, Ta be an α1 ce-set in H, for each a ∈ A, and S = ⋃ a∈A({a} × Ta). Then S is a distance 1-cost effective set in G[H]. Proof: Let A be an α1 ce-set in G, Ta be an α1 ce-set in H, for each a ∈ A, and S =⋃ a∈A({a}×Ta). Let (u, v) ∈ S. Then |N1 G[H](u, v)∩S| = |N1 Hu (v)∩Tu|+ |N1 G(u)∩A||Tu| and |N1 G[H](u, v)∩(V (G[H]) \ S)| = |N1 Hu (v) ∩ (V (H) \ Tu)|+ |N1 G(u) ∩ (V (G) \A)||V (H)| + |N1 G(u) ∩A||V (H) \ Tu|. Hence, |N1 G[H](u, v) ∩ (V (G[H]) \ S)| − |N1 G[H](u, v) ∩ S| = |N1 Hu (v) ∩ (V (H) \ Tu)| − |N1 Hu (v) ∩ Tu|+ |N1 G(u) ∩ (V (G) \A)||V (H)| − |N1 G(u) ∩A||Tu|+ |N1 G(u) ∩A||V (H) \ Tu| ≥ |N1 Hu (v) ∩ (V (H) \ Tu)| − |N1 Hu (v) ∩ Tu|+ |N1 G(u) ∩ (V (G) \A)||V (H)| − |N1 G(u) ∩A||V (H)| +|N1 G(u) ∩A||V (H)| − |N1 G(u) ∩A||V (H)| = |N1 H(v) ∩ (V (Hu) \ Tu)| − |N1 Hu (v) ∩ Tu| + [ |N1 G(u) ∩ (V (G) \A)| − |N1 G(u) ∩A| ] |V (H)|. Since Tu is an α1 ce-set in Hu,∀ u ∈ A , |N1 Hu (v) ∩ (V (H) \ Tu)| − |N1 Hu (v) ∩ Tu| ≥ 0. Also, since A is an α1 ce-set in G, |N1 G(u) ∩ (V (G) \A)| − |N1 G(u) ∩A| ≥ 0. Thus, |N1 G[H](u, v) ∩ (V (G[H]) \ S)| − |N1 G[H](u, v) ∩ S| ≥ 0. Therefore, S is a distance 1-cost effective in G[H]. Corollary 5. Let G and H be connected graphs. Then α1 ce(G[H]) ≥ α1 ce(G)α1 ce(H). Julius G. Caadan, Rolando N. Paluga, Imelda S. Aniversario / Eur. J. Pure Appl. Math, 16 (1) (2023), 261-270 268 Corollary 6. Let G and H be connected graphs. Let A and B be α1 ce-sets in G and H, respectively. Then A×B is a distance 1-cost effective set in G[H]. Definition 1. LetG be a nontrivial connected graph and k ≥ 1 be an integer. A nonempty set S ⊆ V (G) is said to be a very distance k-cost effective set in G if for every u ∈ S, |Nk G(u) ∩ Sc| − |Nk G(u) ∩ S| > 0. Example 1. Consider the graph G as shown in Figure 1. For each v ∈ S = {v2, v3, v4, v5} |N2 G(v) ∩ (V (G) \ S)| − |N2 G(v) ∩ S| > 0. Thus, S is a very distance 2-cost effective set in G. v1 v2 v5 v3 v4 v7 v8 v6 Figure 1: The Graph G with a very distance 2-cost effective set in G Theorem 6. Let G and H be connected graphs and k ≥ 1 be an integer. If A is a very distance k-cost effective set in G, then A×V (H) is a distance k-cost effective set in G[H]. Proof: Let A be a very distance k-cost effective set in G and k ≥ 2 be a positive integer. Let (u, v) ∈ A× V (H). Then∣∣Nk G[H](u, v) ∩ (A× V (H)) ∣∣ = |Nk H(v)|+ |Nk G(u) ∩A||V (H)| = |Nk G(u) ∩A||V (H)|+ |V (H)|. and ∣∣∣Nk G[H](u, v) ∩ ( V (G[H]) \ (A× V (H)) )∣∣∣ = |Nk G(u) ∩ (V (G) \A)||V (H)|. Hence, |Nk G[H](u, v) ∩ (V (G[H])\(A× V (H)))| − |Nk G[H](u, v) ∩ (A× V (H))| = |Nk G(u) ∩ (V (G) \A)||V (H)| − |Nk G(u) ∩A||V (H)| − |V (H)| = ( |Nk G(u) ∩ (V (G) \A)| − |Nk G(u) ∩A| − 1 ) |V (H)|. Since A is a very distance k-cost effective set in G, |Nk G(u)∩(V (G)\A)|−|Nk G(u)∩A| > 0. Thus, ∣∣Nk G[H](u, v)∩ ( V (G[H])\(A×V (H)) )∣∣−|Nk G[H](u, v)∩(A×V (H))| ≥ 0. Accordingly, A× V (H) is a distance k-cost effective set in G[H]. REFERENCES 269 Now for k = 1, let A be a very distance 1-cost effective set in G. Then for each (u, v) ∈ A× V (H), we have |N1 G[H](u, v) ∩ (V (G[H])\(A× V (H)))| − |N1 G[H](u, v) ∩ (A× V (H))| = |N1 G(u) ∩ (V (G) \A)||V (H)| − |N1 G(u) ∩A||V (H)| − |N1 H(v)| = ( |N1 G(u) ∩ (V (G) \A)| − |N1 G(u) ∩A| ) |V (H)| − |N1 H(v)| ≥ ( |N1 G(u) ∩ (V (G) \A)| − |N1 G(u) ∩A| ) |V (H)| − |V (H)| = ( |N1 G(u) ∩ (V (G) \A)| − |N1 G(u) ∩A| − 1 ) |V (H)|. Since A is a very distance 1-cost effective set in G, |N1 G(u)∩ (V (G)\A)|−|N1 G(u)∩A| > 0. Thus, |N1 G[H](u, v) ∩ ( V (G[H]) \ (A× V (H)) ) | − |N1 G[H](u, v) ∩ (A× V (H))| ≥ 0. Hence, A × V (H) is a distance 1-cost effective set in G[H]. Therefore, A × V (H) is a distance k-cost effective set in G[H]. Corollary 7. Let G and H be connected graphs and k ≥ 1 be an integer. Then αk ce(G[H]) ≥ αk ce(G)|V (H)|. 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