Ratio Mathematica Volume 47, 2023 Equitable eccentric domination in graphs Riyaz Ur Rehman A* A Mohamed Ismayil† Abstract In this paper, we define equitable eccentric domination in graphs. An eccentric dominating set S ⊆ V (G) of a graph G(V, E) is called an equitable eccentric dominating set if for every v ∈ V − S there ex- ist at least one vertex u ∈ V such that |d(v) − d(u)| ≤ 1 where vu ∈ E(G). We find equitable eccentric domination number γeqed(G) for most popular known graphs. Theorems related to γeqed(G) have been stated and proved. Keywords: eccentricity, equitable domination number, equitable ec- centric domination number. 2020 AMS subject classifications: 05C69. 1 *Research scholar, PG & Research Department of Mathematics, Jamal Mohamed College (Af- filiated to Bharathidasan University), Tiruchirappalli, India; Mail Id: fouzanriyaz@gmail.com. †Associate Professor, PG & Research Department of Mathematics, Jamal Mohamed College (Affiliated to Bharathidasan University), Tiruchirappalli, India; Mail Id: amis- mayil1973@yahoo.co.in. 1Received on September 15, 2022. Accepted on December 15, 2022. Published online on January 10, 2023. DOI: 10.23755/rm.v41i0.802. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper is published under the CC-BY licence agreement. 87 Riyaz Ur Rehman A and A Mohamed Ismayil 1 Introduction A graph is a representation of a pair of sets (V, E), where V is the set of vertices and E is the set of edges which are connecting the pair of vertices. Graph theory has its application in many fields such as computation, social and natural science etc. Any problems of mathematics, science and engineering can be represented in the form of a graph. The concept of graph theory was first introduced by Leonard Euler in the year 1736. He created the first graph as a solution to solve the prob- lem of seven bridges of Konigsberge built across the pregel river of prussia. Graph theory has experienced tremendous growth, the main reason for this phenomena is applicability of graph theory in different disciplines. Graph theory becomes in- teresting because graphs can be used to model situations that occur in real world problems. These problems can be studied with the aid of graphs. The concept of domination in graphs was studied by Ore and Berge. Ore[11] in- troduced domination in graphs in his famous book ’Theory of graphs’ in 1962. Cockyane and Hedetniemi[3] also contributed several results pertaining to domi- nation. They unfolded different aspects, by swaying all available results bringing to light new ideas and emphasizing its applicable potential in a variety of scien- tific ideas in their paper ’Towards a theory of domination in graphs’. T.W.Haynes, S.Hedetniemi and P.Slater[6] have breifly discussed on various domination pa- rameters in the book Fundamentals of domination in graphs. T.N. Janakiraman et al[9] introduced the concept of eccentric domination in graphs in 2010. Kuppusamy Markandan Dharmalingam[4] introduced equitable graph of a graph. E. Sampathkumar et al[1] introduced degree equitable sets in a graph. V Swaminathan and K.M. Dharmalingam[12] introduced degree equitable domina- tion in graphs. Basavanagoud et al[2] introduced equitable dominating graph. The concept of eccentricity by T.N. Janakiraman et al has inspired researchers which has led to many invariants of eccentric dominations in graphs. Some of the extended eccentric dominations are accurate eccentric domination[7] and equal eccentric domination[8]. The concept of geodesic distance is very important. The existing eccentric domination only highlighted the idea based on an eccentric ver- tex and its domination. The proposed equitable eccentric domination was mainly necessary because it highlights the properties of a vertex in a graph, it considers the connectivity between the vertices where the difference between their vertex de- grees is less than or equal to one. Equitable domination when incorporated with eccentric domination yeilds equitable eccentric domination which concentrates on the vertex degree, geodesic distance, eccentricity, eccentric vertex and domi- nation. In this paper, we introduce equitable eccentric domination in graphs. We 88 Equitable eccentric domination find equitable eccentric dominating set, equitable eccentric domination number γeqed(G), upper equitable eccentric dominating set and upper equitable eccentric domination number Γeqed(G) of different standard graphs. For undefined graph terminologies refer the book ’Graph theory’ by frank harary[5]. 2 Preliminaries Definition 2.1 (11). Let G be a graph with the vertex set V . A subset D of V is a dominating set for G when every vertex not in D is the endpoint of some edge from a vertex in D. Definition 2.2 ([10]). Let γ(G) (called the domination number) and Γ(G) (called the upper domination number) be the minimum cardinality and the maximum car- dinality of a minimal dominating set of G, respectively. Definition 2.3 ([6]). The degree deg(v) of v is the number of edges incident with v. Definition 2.4 ([9]). The eccentricity e(v) of v is the distance to a vertex farthest from v. Thus, e(v) = max{d(u, v) : u ∈ V }. For a vertex v, each vertex at a distance e(v) from v is an eccentric vertex. Eccentric set of a vertex v is defined as E(v) = {u ∈ V (G)/d(u, v) = e(v)}. Definition 2.5 ([9]). The radius r(G) is the minimum eccentricity of the vertices, whereas the diameter diam(G) is the maximum eccentricity. Definition 2.6 ([9]). v is a central vertex if e(v) = r(G). The center C(G) is the set of all central vertices. v is a peripheral vertex if e(v) = diam(G). The periphery P(G) is the set of all peripheral vertices. Definition 2.7 ([9]). A set D ⊆ V (G) is an eccentric dominating set if D is a dominating set of G and for every v ∈ V − D, there exists at least one eccentric point of v in D. If D is an eccentric dominating set, then every superset D′ ⊇ D is also an eccentric dominating set. But D′′ ⊆ D is not necessarily an eccentric dominating set. An eccentric dominating set D is a minimal eccentric dominating set if no proper subset D′′ ⊆ D is an eccentric dominating set. Definition 2.8 ([9]). The eccentric domination number γed(G) of a graph G equals the minimum cardinality of an eccentric dominating set. That is, γed(G) = min|D|, where the minimum is taken over D in D, where D is the set of all minimal eccen- tric dominating sets of G. Definition 2.9 ([4]). A subset D of V is called an equitable dominating set if for every v ∈ V − D there exists a vertex u ∈ D such that uv ∈ E(G) and |deg(u) − deg(v)| ≤ 1. The minimum cardinality of such a dominating set is denoted by γe and is called the equitable domination number of G. 89 Riyaz Ur Rehman A and A Mohamed Ismayil 3 Equitable eccentric domination in graphs In this section we introduce equitable eccentric domination, theorems related to equitable eccentric domination number of family of graphs are stated and proved. Definition 3.1. An eccentric dominating set S ⊆ V (G) is called an equitable eccentric dominating set(EQED-set) if for every v ∈ V − S there exist at least one vertex u ∈ S such that vu ∈ E(G) and |d(v) − d(u)| ≤ 1. Definition 3.2. An equitable eccentric dominating set S is called a minimal eq- uitable eccentric dominating set if no proper subset of S is equitable eccentric dominating set. Definition 3.3. The equitable eccentric domination number γeqed(G) of a graph G is the minimum cardinality among the minimal equitable eccentric dominating sets of G. Definition 3.4. The upper equitable eccentric domination number Γeqed(G) of a graph G is the maximum cardinality among the minimal equitable eccentric dominating sets of G. Example 3.1. v4 v5 v2 v3 v6 v1 Figure 2.1: Graph G Consider the graph G consists of 6 vertices given in figure 2.1. Here the dominat- ing set is S = {v1, v4} but not eccentric dominating set since E(v3) = {v2, v6} not in S. The eccentric dominating set is S = {v1, v6} but not equitable eccentric dominating set since |d(v4) − d(v6)| = 2. The equitable eccentric dominating set is S = {v1, v2, v6}. Remark 3.1. For any path Pn where n ≥ 3, 1. Every minimum EQED-set contains the pendant vertices. 2. If D1, D2, D3 are minimum EQED-sets of paths Pn−1, Pn, Pn+1 consecutively where n = 3k and k > 1. Then |D1| = |D2| = |D3|. Therefore for k = 2, γeqed(P5) = γeqed(P6) = γeqed(P7) = 3. Theorem 3.1. For complete graph Kn, γeqed(Kn) = 1, ∀ n ≥ 2. 90 Equitable eccentric domination Proof. In a complete graph Kn all the vertices are eccentric vertices to each other. If v ∈ V (Kn) then the eccentric vertex E(v) = V (Kn) − {v} and every singleton set forms a dominating set. For every vertex v ∈ D ∃ a vertex u ∈ V (Kn) − D ∋ |deg(u) − deg(v)| ≤ 1 where uv ∈ E(Kn). Therefore every single vertex of Kn is an EQED-set. Hence γeqed(Kn) = 1. Theorem 3.2. For path graph Pn where n > 1, γeqed(Pn) = { 1, for n = 2 ⌊n+1 3 ⌋ + 1, ∀ n ≥ 3 Proof. Case(i): For a path P2, V (P2) = {v1, v2}. Both the vertices are eccen- tric vertices to each other. Therefore D = {v1} or {v2} and |deg(v2)−deg(v1)| = 0, where v1v2 ∈ E(P2). Hence γeqed(P2) = 1. Case(ii): For a path Pn where n ≥ 3. The pendant or end vertices of the path form the eccentric vertices ie, if V (Pn) = {v1, v2, v3, . . . vn}, E(v1) = {vn} and E(vn) = {v1}. E(vi) = {v1} or {vn} for any vi ∈ V (Pn) where n is even. If n is odd then E(vi) = {v1} or {vn}. For Pn where ′n′ is odd, the central vertex vi has two eccentric vertices ie, E(vi) = {v1, vn}. Degree of end vertices is 1 and de- gree of all the intermediate vertices is 2. The EQED-set contains both the pendant vertices. Both v1 and vn being pendant vertices dominate the vertices adjacent to them and the minimum dominating set among the intermediate vertices along with two pendant vertices forms an EQED-set. Since |deg(u)−deg(v)| ≤ 1 where uv ∈ E(Pn) for all u ∈ D and v ∈ V (Pn) − D and ∃ an eccentric vertex u ∈ D for every v ∈ V (Pn) − D. For Pn where n = 3k and k > 2, number of vertices of P3k−1, P3k, P3k+1 are same. Every minimum equitable eccentric domination set of D contains ⌊n+1 3 ⌋ + 1 number of vertices. Theorem 3.3. For star graph Sn, γeqed(Sn) = { 2, if n = 3 0, if n ̸= 3 Proof. Case(i): If n = 3, then the star graph S3 is isometric to P3. From the theorem-3.2 γeqed(P3) = γeqed(S3) = 2. Case(ii): If n ̸= 3 then Sn is of the form S4, S5, S6, . . . For any graph Sn where n ̸= 3, there can be many dominating sets and eccentric dominating sets but we cannot find a EQED-set because of the central vertex vi of the star graph has degree ≥ 3. The degree of every pendant vertex u of a star graph is 1, deg(u) = 1, u ∈ V (Sn) − {vi}. The degree of central vertex vi of a star graph is given by deg(vi) = n − 1. Since, central vertex vi ∈ V (Sn) then either vi ∈ D or vi ∈ V (Sn) − D. Therefore |deg(vi) − deg(u)| > 1 always which doesnot satisfy the condition to be a EQED-set. Hence γeqed(Sn) = 0 where n ̸= 3. 91 Riyaz Ur Rehman A and A Mohamed Ismayil Theorem 3.4. For cycle graph Cn where n ≥ 3, γeqed(Cn) =   n 2 , if n is even ∀ n ≥ 4 ⌈n 3 ⌉, if n is odd & n = 3k ∀ k = 1, 3, 5, 7, . . . ⌈n 3 ⌉ + 1, otherwise Proof. Case(i): If ′n′ is even and n ≥ 4. Let the cycles Cn be of the form C4, C6, C8, C10, . . . C2n. In an even cycle if u ∈ V (Cn) the eccentric vertex of u, E(u) = {v} is always placed at a distance of n 2 edges from it and every vertex has a unique eccentric vertex to form the first eccentric dominating set. The set D must contain n 2 vertices in such a way that for every v ∈ D then E(v) /∈ D or for some u ∈ V (Cn) − D, E(u) /∈ V (Cn) − D. Then if the vertex u and E(u) ∈ D then we cannot construct a eccentric dominating set. Fur- ther if we reduce the cardinality of D to less than n 2 we will have u and E(u) in V − D. Therefore D must contain n 2 vertices with all the unique eccentric vertices in V − D. Then for any u ∈ V (Cn) − D ∃ a vertex v ∈ D such that |deg(u) − deg(v)| ≤ 1 where uv ∈ E(Cn) for every vertex vi ∈ Cn, deg(vi) = 2. Therefore |deg(u) − deg(v)| = |2 − 2| = 0. Therefore γeqed(Cn) = n2 . Case(ii): Now we have the odd cycles of the form C3, C9, C15, C21, . . . C3k. Every vertex u ∈ V (Cn) has two eccentric vertices vi, vj such that E(u) = {vi, vj}. The eccentric vertices vi, vj will always be adjacent i.e., vi, vj ∈ E(Cn). vi, vj are placed at a distance of n−1 2 edges from u. Since every vertex u can dom- inate its adjacent vertices v, w. n 3 set of vertices form a dominating set of a cycle. The dominating set D = {vi, vj, vn} forms the EQED-set such that no eccentric vertices of vi ∈ D are in D. Then ∀ v ∈ V (Cn) − D ∃ a vertex u ∈ D ∋′ |deg(u) − deg(v)| = |2 − 2| = 0. Therefore γeqed(Cn) = ⌈n3 ⌉. Case(iii): If n = 3k+1 where k is even. The cycles are of the form C7,C13,C19,. . . , C3k+1 and if n = 3k+1 where k is odd, the cycles are of the form C5,C11,C17,. . . , C3k+2. Totally we have C5, C7, C11, C13, C17, C19, . . . C3k+1, C3k+2. Similar to case(ii) every vertex vi ∈ V (Cn) has two eccentric vertices vl, vm, E(vi) = {vl, vm} such that vl and vm are adjacent i.e., vl, vm ∈ E(Cn). Eccentric ver- tex vl and vm of vi are placed at a distance of n−1 2 from vi. If n = 3k we get 3, 9, 15, 21, . . . which are the multiples of 3 we get a whole number which forms the cardinality of a EQED-set as proved in case(ii). But when n = 3k + 1 or n = 3k +2 then n = 5, 7, 11, 13, 17, 19, . . . 3k +1, 3k +2 which are not multiples of 3 we get a fraction value and also we are left out with a vertex which is to be dominated. Therefore the cardinality of the EQED-set of a cycles of the form C3k+1, C3k+2 increases by 1. Hence γeqed(Cn) = ⌈n3 ⌉ + 1. Theorem 3.5. Every EQED-set in a wheel graph Wn, n ≥ 6 contains the central vertex. 92 Equitable eccentric domination Proof. Let v1 be the central vertex of the wheel graph Wn, n ≥ 6 then deg(v1) = n − 1 = ∆(Wn). The degree of any non-central vertex u ∈ V (Wn) is deg(u) = 3 = δ(Wn). Suppose the central vertex v1 ∈ V (Wn) − D, u ∈ D and D is an minimal ec- centric dominating set we need to check for the condition of equitable domination then for v1 ∈ V (Wn) − D and u ∈ D, we have uv1 ∈ E(Wn) |deg(v1) − deg(u)| = |∆(G) − δ(G)| |deg(v1) − deg(u)| = |(n − 1) − 3| |deg(v1) − deg(u)| = |n − 4| where n ≥ 6 |deg(v1) − deg(u)| > 1. which is a contradiction. Therefore the central vertex v1 must belong to D, if the set D is a equitable ec- centric dominating set of Wn. Theorem 3.6. Let Wn be a wheel graph where n ≥ 5 then EQED-set contains more than one vertex. Proof. In any wheel graph Wn where n ≥ 5. If the set D ⊆ V (Wn) contains the central vertex v1 then D forms a dominating set as deg(v1) = n − 1 = ∆(G). But the eccentric vertices of a central vertex v1 is given by E(v1) = V − {v1} and the eccentric vertex of any non-central vertex u is given by E(u) = V − N[u]. Therefore there is no eccentric dominating or equitable eccentric dominating set of cardinality 1 for Wn where n ≥ 5. Theorem 3.7. For wheel graph Wn, where n ≥ 4 we have γeqed(Wn) =   1, if n = 4 4, if n = 6 ⌊n 2 ⌋, if n is odd and n ≥ 5 ⌊n+1 3 ⌋ + 1, ∀ n ≥ 8 and n is even Proof. Case(i): If n = 4, W4 is isometric to K4, then by theorem-3.1 γeqed(W4) = γeqed(K4) = 1. Case(ii): If n = 6, in a wheel graph W6, there are no eccentric dominating sets of cardinality 1 or 2. Therefore we do not get an EQED-set of cardinality 1 or 2. There are sets of cardinality 3 which are eccentric dominating sets. But they do not form an EQED-set as the central vertex should not be present in V −D. Since the degree of central vertex vi is deg(vi) = n − 1 = 5 and degree of any other non-central vertex is deg(vj) = 3. Therefore |deg(vi) − deg(vj)| = 2 > 1 and in other cases if vi /∈ V − D then we find a combination of vertices of 3 cardinal- ity which are eccentric dominating set but they dont form an INED-set since for some vertex v ∈ V − D there is no vertex u ∈ D such that u, v /∈ E(W6). But we find a EQED-set with cardinality 4 as we have the central vertex in D. Then |deg(vi) − deg(vj)| ≤ 1, (vi, vj) ∈ E(W6) where vi ∈ D and vj ∈ V (W6) − D. 93 Riyaz Ur Rehman A and A Mohamed Ismayil Therefore γeqed(W6) = 4. Case(iii): If n is odd and n ≥ 5 we have the wheel graph of order W5, W7, W9, W11, . . . If v ∈ V (Wn) then |E(v)| = n − 4. There will always be n − 4 vertices which form the eccentric vertex E(v) for every vertex v. And for any wheel graph where ′n′ is odd. The set D ⊆ V (Wn) forms an eccentric dominating sets only when |D| = ⌊n 2 ⌋. Then for every v ∈ V (Wn) − D there exists a vertex u ∈ D such that |deg(u) − deg(v)| ≤ 1 and (u, v) ∈ E(Wn). Therefore γeqed(Wn) = ⌊n2 ⌋. Case(iv): The wheel graph Wn where n is even and n ≥ 8 has n − 4 eccentric vertices. We have wheel graphs W8, W10, W12, . . . For every vertex v ∈ V (Wn), |E(v)| = n − 4. From theorem-3.6,3.5, γeqed(Wn) ̸= 1 and the central vertex vi ∈ D then D contains other vertices of Wn where cardinality of D is of the form ⌊n+1 3 ⌋ + 1. For every v ∈ V − D there exists a vertex u ∈ D such that E(v) lies in D and |deg(u) − deg(v)| ≤ 1 such that there exists an edge between u and v. Therefore γeqed(Wn) = ⌊n+13 ⌋ + 1. Theorem 3.8. An EQED-set D is a minimal EQED-set if one of the following conditions holds, 1. For every vertex u in V − D there does not exists v in D such that E(u) = {v} ie, u has no eccentric vertex in D. 2. There exists some u ∈ V − D such that N(u) ⋂ D = {v}, E(u) ⋂ D = {v} and |d(u) − d(v)| ≤ 1 where uv ∈ E(G). Proof. Suppose D is a minimal EQED-set of G. Then for every vertex v in D, D − {v} is not an EQED-set. Thus there exists some vertex u in V − D ⋃ {v} which is not dominated by any vertex in D −{v} or there exists u ∈ V −D ⋃ {v} such that u does not have an eccentric vertex in D − {v} ie, E(u) ̸= D − {v} or |d(u) − d(v)| ≰ 1 or uv /∈ E(G). ∴ The concept of equitable condition does not hold. Case(i): If v = u then u does not have an eccentric vertex in D ie, E(u) ̸= D. Case(ii): If v ̸= u, (a) If u ∈ V −D and u is not dominated by D−{v}, but dominated by D then u is adjacent to only v in D ie,N(u) ⋂ D = {v}. (b) If u ∈ V −D and u does not have an eccentric vertex in D−{v} but u has an eccen- tric vertex in D. Thus v is the only eccentric vertex of u in D ie, E(u) ⋂ D = {v}. (c) If u ∈ V − D and |d(u) − d(x)| ≰ 1 or ux /∈ E(G) where x ∈ D − {v} but |d(u) − d(v)| ≤ 1 and uv ∈ E(G). Conversely, Suppose D is an EQED-set and for each v ∈ D, one of the two conditions holds. Now we show that D is a min- imal EQED-set. Suppose D is not an minimal EQED-set ie, there exists a vertex v ∈ D such that D − {v} is an EQED-set. Hence v is adjacent to at least one vertex x in D −{v}, v has an eccentric vertex in D −{v} ie, E(v) ∈ D −{v} and |d(u) − d(x)| ≤ 1 where ux ∈ E(G). ∴ Equitable condition holds and EQED-set exists. Also if D − {v} is an EQED-set, then every vertex u in V − D is ad- jacent to at least one vertex x in D − {v}, u has an eccentric vertex in D − {v} ie,E(u) ∈ D−{v} and |d(u)−d(x)| ≤ 1 and ux ∈ E(G). Therefore condition-(2) 94 Equitable eccentric domination does not hold. Hence neither condition-(1) nor (2) holds, which is a contradiction to our assumption. Hence for each v ∈ D one of the 2 conditions holds. The equitable eccentric dominating set, γeqed(G), upper equitable eccentric dom- inating set and Γeqed(G) of standard graphs are tabulated. Graph Figure D - Minimum EQED set. |D| = γeqed(G) γeqed(G) S - Upper EQED set. |S| = Γeqed(G) Γeqed(G) Diamond graph v1 v4 v2 v3 {v1, v2}, {v1, v3}, {v2, v3}, {v2, v4}, {v3, v4}. 2 {v1, v2}, {v1, v3}, {v2, v3}, {v2, v4}, {v3, v4}. 2 Tetrahedral graph v2 v1 v3 v4 {v1}, {v2}, {v3}, {v4}. 1 {v1}, {v2}, {v3}, {v4}. 1 Claw graph v2 v3 v1 v4 Does not exist 0 Does not exist 0 Paw graph v2 v3 v1 v4 {v1, v3}, {v2, v3}, {v3, v4}. 2 {v1, v3}, {v2, v3}, {v3, v4}. 2 Bull graph v3 v4 v5 v2v1 {v1, v2, v3}, {v1, v2, v4}, {v1, v2, v5}. 3 {v1, v2, v3}, {v1, v2, v4}, {v1, v2, v5}. 3 Butterfly graph v3 v2 v5 v1 v4 {v1, v2, v3}, {v1, v3, v5}, {v2, v3, v4}, {v3, v4, v5}. 3 {v1, v2, v3}, {v1, v3, v5}, {v2, v3, v4}, {v3, v4, v5}. 3 Banner graph v3 v4 v1 v2 v5 {v1, v2, v5}, {v1, v3, v5}, {v2, v3, v5}, {v2, v4, v5}, {v3, v4, v5}. 3 {v1, v2, v5}, {v1, v3, v5}, {v2, v3, v5}, {v2, v4, v5}, {v3, v4, v5}. 3 95 Riyaz Ur Rehman A and A Mohamed Ismayil Graph Figure D - Minimum EQED set. |D| = γeqed(G) γeqed(G) S - Upper EQED set. |S| = Γeqed(G) Γeqed(G) Fork graph v2 v3 v1 v4 v5 {v1, v2, v3, v4}, {v1, v2, v4, v5}, {v1, v3, v4, v5}. 4 {v1, v2, v3, v4}, {v1, v2, v4, v5}, {v1, v3, v4, v5}. 4 (3,2)-Tadpole graph v2 v3 v4 v1 v5 {v1, v4}, {v4, v5}. 2 {v1, v2, v3, v5}. 4 Kite graph v3 v4 v1 v5 v2 {v1, v2, v4}, {v1, v3, v4}, {v2, v3, v4}, {v2, v4, v5}, {v3, v4, v5}. 3 {v1, v2, v4}, {v1, v3, v4}, {v2, v3, v4}, {v2, v4, v5}, {v3, v4, v5}. 3 (4,1)-Lollipop graph v3 v4 v1 v5 v2 {v1, v4}, {v2, v4}, {v3, v4}, {v4, v5}. 2 {v1, v4}, {v2, v4}, {v3, v4}, {v4, v5}. 2 House graph v2 v3 v1 v4 v5 {v2, v4}, {v3, v5}. 2 {v1, v2, v3}, {v1, v4, v5}. 3 House X graph v2 v3 v1 v4 v5 {v1, v2}, {v1, v3}, {v1, v4}, {v1, v5}. 2 {v1, v2}, {v1, v3}, {v1, v4}, {v1, v5}. 2 Gem graph v1 v2 v5 v3 v4 {v1, v2}. 2 {v1, v3, v4}, {v2, v3, v4}, {v3, v4, v5}. 3 96 Equitable eccentric domination Graph Figure D - Minimum EQED set. |D| = γeqed(G) γeqed(G) S - Upper EQED set. |S| = Γeqed(G) Γeqed(G) Dart graph v3 v4 v1 v5 v2 {v2, v4}. 2 {v1, v2, v3, v5}. 4 Cricket graph v4 v5v3 v1 v2 {v1, v3, v4, v5}, {v2, v3, v4, v5}. 4 {v1, v3, v4, v5}, {v2, v3, v4, v5}. 4 Pentatope graph v1 v4 v5 v2 v3 {v1}, {v2}, {v3}, {v4}, {v5}. 1 {v1}, {v2}, {v3}, {v4}, {v5}. 1 Johnson solid skeleton-12 graph v2 v1 v3 v4 v5 {v1, v2}, {v1, v3}, {v1, v4}, {v1, v5}, {v2, v3}, {v3, v4}, {v3, v5}. 2 {v1, v2}, {v1, v3}, {v1, v4}, {v1, v5}, {v2, v3}, {v3, v4}, {v3, v5}. 2 Cross graph v3 v1 v2 v4 v5 v6 {v1, v2, v3, v4, v5}, {v1, v2, v3, v4, v6}. 5 {v1, v2, v3, v4, v5}, {v1, v2, v3, v4, v6}. 5 Net graph v5 v6 v3 v4 v1 v2 {v1, v2, v3, v6}, {v1, v2, v4, v6}, {v1, v2, v5, v6}. 4 {v1, v2, v3, v6}, {v1, v2, v4, v6}, {v1, v2, v5, v6}. 4 Fish graph v4 v2 v5 v1 v6 v3 {v2, v3, v4}, {v3, v4, v5}. 3 {v1, v2, v4, v5, v6}. 5 97 Riyaz Ur Rehman A and A Mohamed Ismayil Graph Figure D - Minimum EQED set. |D| = γeqed(G) γeqed(G) S - Upper EQED set. |S| = Γeqed(G) Γeqed(G) A graph v3 v4 v1 v5 v2 v6 {v1, v2, v5, v6}, {v1, v3, v5, v6}, {v1, v4, v5, v6}, {v2, v3, v5, v6}, {v2, v4, v5, v6}, {v3, v4, v5, v6}. 4 {v1, v2, v5, v6}, {v1, v3, v5, v6}, {v1, v4, v5, v6}, {v2, v3, v5, v6}, {v2, v4, v5, v6}, {v3, v4, v5, v6}. 4 R graph v3 v4 v1 v5 v2 v6 {v2, v3, v5, v6}. 4 {v1, v3, v4, v5, v6}. 5 4-polynomial graph v2 v3v1 v5v4 v6 {v1, v2, v3}, {v1, v3, v4}, {v2, v3, v4}, {v3, v4, v5}, {v3, v4, v6}, {v4, v5, v6}. 3 {v1, v2, v3}, {v1, v3, v4}, {v2, v3, v4}, {v3, v4, v5}, {v3, v4, v6}, {v4, v5, v6}. 3 (2,3)-King graph v2 v3v1 v5v4 v6 {v1, v2, v3}, {v1, v2, v6}, {v1, v3, v5}, {v1, v5, v6}, {v2, v3, v4}, {v2, v4, v6}, {v3, v4, v5}, {v4, v5, v6}. 3 {v1, v2, v3}, {v1, v2, v6}, {v1, v3, v5}, {v1, v5, v6}, {v2, v3, v4}, {v2, v4, v6}, {v3, v4, v5}, {v4, v5, v6}. 3 Antenna graph v2 v1 v3 v4 v5 v6 {v1, v2, v5}, {v1, v2, v6}, {v1, v3, v5}, {v1, v3, v6}, {v1, v4, v5}, {v1, v4, v6}. 3 {v1, v2, v3, v4}. 4 3-prism graph v2 v3 v4 v1 v5 v6 {v1, v2}, {v3, v5}, {v4, v6}. 2 {v1, v5, v6}, {v2, v3, v4}. 3 Octahedral graph v4 v3v2 v1 v5 v6 {v1, v2, v3}, {v1, v2, v5}, {v1, v3, v6}, {v1, v5, v6}, {v2, v3, v4}, {v2, v4, v5}, {v3, v4, v6}, {v4, v5, v6}. 3 {v1, v2, v3}, {v1, v2, v5}, {v1, v3, v6}, {v1, v5, v6}, {v2, v3, v4}, {v2, v4, v5}, {v3, v4, v6}, {v4, v5, v6}. 3 98 Equitable eccentric domination 4 Conclusions Inspired by eccentric dominating set and equitable dominating set we introduce the equitable eccentric dominating set. 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