We introduce transversal eccentric dominating(TED) set. An eccentric dominating set $D$ is called a TED-set if it intersects with every minimum eccentric dominating set $D'$. We find the TED-number $\gamma_{ted}$ of some standard graphs. Results are stated and proved

Ismayil, A Mohamed

Rehman A, Riyaz Ur

English

APAV - Academy of Sciences, Letters, Arts and Technology (E-Journals)

Ratio Mathematica Volume 46, 2023Transversal eccentric domination in graphsRiyaz Ur Rehman A*A Mohamed Ismayil†AbstractEccentricity of a vertex vis a maximum among the shortest distancesbetween the vertex v and all other vertices. A set Dis called eccentricdominating if every vertex in its compliment has an eccentric ver-tex in the set D.A dominating set is transversal if the intersectionof the set with all the minimum dominating sets is non-empty. In-spired by both the concepts we introduce transversal eccentric domi-nating(TED) set. An eccentric dominating set D is called a TED-setif it intersects with every minimum eccentric dominating set D’. Wefind the TED-number γted (G) of family of graphs, theorems relatedto their properties are stated and proved.Keywords: Eccentricity, TED-set, TED-number.2020 AMS subject classifications: 05C69. 1*Jamal Mohamed College (Affiliated to Bharathidasan University), Tiruchirappalli, India.fouzanriyaz@gmail.com.†Jamal Mohamed College (Affiliated to Bharathidasan University), Tiruchirappalli, India.amismayil1973@yahoo.co.in.1Received on September 15, 2022. Accepted on December 15, 2022. Published on March 20,2023. DOI: 10.23755/rm.v46i0.1043. ISSN: 1592-7415. eISSN: 2282-8214. ©Riyaz Ur Rehmanet al. This paper is published under the CC-BY licence agreement.24Riyaz Ur Rehman A and A Mohamed Ismayil1 IntroductionThe classical queens problem in chess or the study of networks in electronicsdomination finds its application everywhere and plays a pivotal role in modernday science and technology. Domination is a vast arena in graph theory which isjust not limited to adjacency between vertices belonging to the dominating set andits compliment. For a graph G(V,E), a set S ⊆ V is said to be a dominating set,if every vertex in V-S is adjacent to some vertex in S. The domination number γd(G) of a graph G equals the minimum cardinality of an dominating set. There aremany different invariants of domination. The concept of transversal domination ingraphs was introduced by Nayaka S.R, Anwar Alwardi and Puttaswamy in 2018.A dominating set Dwhich intersects every minimum dominating set in G is calleda transversal dominating set. The minimum cardinality of a transversal dominat-ing set is called the transversal domination number denoted by γtd (G). Geodesicbeing the shortest distance between any two vertices. The concept of shortest pathhas always intrigued the researchers in graph theory, operation research, computerscience and other fields.There are many different types of distances in graphs,one such distance is eccentricity. The concept of eccentricity incorporated with adominating set yields an eccentric dominating set.Eccentric domination was intro-duced by T. N. Janakiraman et al in 2010. The eccentricitye (v) of v is the distanceto a vertex farthest from v. Thus, e(v) = maxd(u, v) : u ∈ V . For a vertex v, eachvertex at a distancee (v) from v is an eccentric vertex. Eccentric set of a vertex vis defined as E(v) = u ∈ (G) : d(u, v) = e(v). A set D ⊆ V (G) is an eccentricdominating set if D is a dominating set of G and for every v ∈ V − D, thereexists at least one eccentric vertex of v in D. The eccentric domination numberγed (G) of a graph G equals the minimum cardinality of an eccentric dominatingset. The main motive of this paper is to hybrid two different types of dominationsand define a new domination variant. Inspired by this idea we combine transver-sal domination with eccentric domination. In this paper, we introduce transversaleccentric domination and calculate the TED-number of different graphs. Resultsrelated to TED-number of family of complete, star, path, cycle and wheel graphsare discussed. The upper TED-set, upper TED-number, lower TED-set and lowerTED-number of different standard graphs are tabulated. For undefined terminolo-gies refer the book graph theory by frank harary.2 Transversal eccentric domination in graphsDefinition 2.1. An eccentric dominating(ED) set S ⊆ V (G) is called a transver-sal eccentric dominating set(TED-set) if it intersects with every minimum ED-setD′ ie S⋂D′ 6= ∅.25Transversal eccentric domination in graphsDefinition 2.2. A TED-set S is called a minimal TED-set if no proper subset of Sis TED-set.Definition 2.3. The TED-number γted(G) of a graphG is the minimum cardinalityamong the minimal TED-sets of G.Definition 2.4. The upper TED-number Γted(G) of a graph G is the maximumcardinality among the minimal TED-sets of G.Example 2.1. .℘2℘3 ℘4℘1℘5 ℘6Figure 2.1: Graph GConsider the above example where the graph G consists of 6 vertices and 9edges.(i) The dominating sets are {℘1, ℘2}, {℘1, ℘3}, {℘1, ℘4}, {℘2, ℘5}, {℘2, ℘6},{℘3, ℘5}, {℘3, ℘6}, {℘4, ℘5}, {℘4, ℘6}.(ii) The minimum ED-sets are {℘1, ℘2}, {℘3, ℘5}, {℘4, ℘6}.(iii) The TED-sets are {℘1, ℘5, ℘6}, {℘2, ℘3, ℘4}.Observation 2.1. For any graph G,1. γ(G) ≤ γed(G) ≤ γted(G) ≤ Γted(G).2. γted(G) ≤ n and Γted(G) ≤ n.3. V (G) is also a TED-set.Theorem 2.1. For complete graph Kn, γted(Kn) = n, ∀n ≥ 2.Proof: Let V (Kn) = {℘1, ℘2, . . . ℘n}. Since deg(℘i) = n− 1 ∀℘i ∈ V (Kn)the eccentric vertex of ℘i is given by E(℘i) = V − {℘i} and every single vertexdominates all other vertices. Since every vertex ℘i ∈ V forms an ED-set of theform D1 = {℘1}, D2 = {℘2}, D3 = {℘3}, . . . Dn = {℘n}. The vertex set V isthe only set which forms a TED-set, since V (Kn)∩Di 6= ∅where i = 1, 2, 3, . . . nand Di is any ED-set.26Riyaz Ur Rehman A and A Mohamed IsmayilTheorem 2.2. For star graph Sn, γted(Sn) = 2 ∀ n ≥ 3.Proof: Let V (Sn) = {℘1, . . . ℘i, . . . ℘n} where deg(℘i) = n − 1 where ℘i isthe central vertex and deg(℘j) = 1 where ℘j is a pendant vertex of star graph Sn.E(℘i) = V − {℘i} and E(℘j) = V − {℘i, ℘j}. The central vertex ℘i forms adominating set {℘i} but it is not an ED-set for any ℘j ∈ V − D, E(℘j) /∈ D.But D = {℘i, ℘j} forms an ED-set, then for S3 we have 3 ED-sets which formsthe minimum ED-sets and for any star graph Sn, ∀ n ≥ 4, we have (n − 1) ED-sets which forms the minimum ED-sets D1 = {℘i, ℘1}, D2 = {℘i, ℘2}, D3 ={℘i, ℘3}, . . . Dn = {℘i, ℘n}. Any minimum ED-set D = {℘i, ℘j} also forms aTED-set, since D ∩ {℘i, ℘j} = {℘i} 6= ∅. Therefore γted(Sn) = 2 ∀n ≥ 3.Theorem 2.3. For path graph Pn, γted(Pn) = bn+13 c+ 1, ∀ n ≥ 2.Proof: Let the vertices of Pn be given by V (Pn) = {℘1, ℘2, . . . ℘n}. Everypath Pn contains two pendant vertices {℘1, ℘n}. For any vertex ℘i ∈ V (Pn) theeccentric vertex of ℘i is E(℘i) = {℘1} or {℘n} where n is even. If n is odd thenE(℘i) = {℘1} or {℘n} but if ℘i is a vertex equidistant from both the pendantvertices then ℘i = ℘n+12, E(℘n+12) = {℘1, ℘n}. For any path Pn, dn3 e set ofvertices can dominate all the vertices of Pn. Similarly a set D whose cardinalityis bn+13c + 1 will eccentric dominate all the vertices of Pn. By the definition ofTED-set, a set D should intersect all the minimum ED-set. An ED-set D willintersect all the minimum ED-sets. Therefore every minimum ED-set is a TED-set. Therefore γed(Pn) = γted = bn+13 c+ 1Theorem 2.4. For cycle graph Cn where n ≥ 3γted(Cn) ={5, for n = 8dn+13e+ 1, otherwiseProof: Case(i): For C8, the set D = {℘i, ℘j, ℘k, ℘l} whose cardinality isdn+13e + 1 = 4 does not form a TED-set which is an exception from case(i).Adding a vertex to D is of the form {℘i, ℘j, ℘k, ℘l, ℘m} whose cardinality is fivewill increasing the cardinality of D. Here every vertex in V (C8) − D has an ec-centric vertex in D and D is also dominating set which intersects all the minimumdominating sets of C8. Therefore γted(C8) = 5.Case(ii): For a cycle graph Cn, if n is even and n 6= 8 then every vertex ℘i ∈V (Cn) has a unique eccentric vertex ie, E(℘i) = {℘j |℘j ∈ V (Cn)}. E(℘i) isat a distance of n2edges from ℘i for an even cycle. If n is odd then every vertex℘i has two eccentric vertices. E(℘i) = {℘j, ℘k |℘j, ℘k ∈ V (Cn)}. E(℘i) is ata distance of bn2c edges from ℘i for odd cycle. Every single vertex ℘i can domi-nate itself and two vertices adjacent to it. Therefore for any cycle Cn, dn3 e set ofvertices forms the dominating set. Here we see that any set D = {℘1, ℘2, . . . ℘i}which has the cardinality of the form dn+13e + 1 forms a dominating set as well27Transversal eccentric domination in graphsas an ED-set. Since D whose cardinality is dn+13e + 1 intersects every minimumED-set of cardinality γed(Cn) ={n2, if n is evendn3e or dn3e+ 1, if n is oddD forms a TED-set. Hence γted(Cn) = dn+13 e+ 1.Theorem 2.5. For wheel graph Wn where n ≥ 4, a ≥ 1γted(Wn) ={3, for n = (6a− 1), (6a) or (6a+ 1)4, for n = (6a− 2), (6a+ 2) or (6a+ 3)Proof: Case(i): If n = 6a − 1, 6a and 6a + 1, the wheel graphs are of theform W5,W6,W7,W11,W12,W13,W17,W18,W19, . . .W6a−1,W6a,W6a+1. Let ℘cbe the central vertex of wheel graph, deg(℘c) = n − 1. Therefore ℘c has n − 1eccentric vertices, |E(℘c)| = n−1. Let ℘i be the non-central vertex, deg(℘i) = 3.Then closed neighbourhood of ℘i ie, N [℘i] = 4. Therefore ℘i has n− 4 eccentricvertices, |E(℘i)| = n− 4. D = {℘c} forms the only dominating set of cardinalityone, but not an ED-set. Other than W5 and W7 every other wheel graph has anED-set D = {℘c, ℘x, ℘y} where ℘c, ℘x, ℘y ∈ V (Wn) forms an ED-set and forevery v ∈ V (Wn)−D there exists a vertex ℘c, ℘x or ℘y in D such that E(v) = ℘cor ℘x or ℘y and D = {℘c, ℘x, ℘y} forms a TED-set, since D intersects everyminimum ED-set. Therefore |D| = 3, γted(Wn) = 3 for n = 6a− 1, 6a, 6a+ 1.Case(ii): If (6a− 2), (6a+ 2) and (6a+ 3), then the wheel graphs are of the formW4,W8,W9,W10,W14,W15,W16, . . .W6a−2,W6a+2,W6a+3. For W4, γted(W4) =4. SinceW4 isK4 which is complete graph (by theorem-2.1). Similar to case(i), ℘cis the central vertex of wheel graph and ℘j is the non-central vertex, |E(℘c)| = n−1 and |E(℘i)| = n−4. Similar to case(i) the only unique dominating setD = {℘c}whose cardinality is one does not form an ED-set. But a set D = {℘c, ℘x, ℘y}containing three vertices forms an ED-set, since every vertex ℘i ∈ V (Wn) − Dhas an eccentric vertex in D ie, E(℘i) = ℘c, ℘x or ℘y. But D = {℘c, ℘x, ℘y}whose cardinality is three does not form a TED-set since it does not intersectevery minimum ED-set. But an addition of vertex ℘z to the same set gives us a setD = {℘c, ℘x, ℘y, ℘z} whose cardinality is four forms an ED-set and it intersectsevery minimum ED-set of cardinality three, thus becoming TED-set. Thereforeγted(Wn) = 4 for n = (6a− 2), (6a+ 2) and (6a+ 3).Proposition 2.1. For any graph G,1. γted(G) ≥ b (2n−q)4 c.2. γted(G) ≥ diam(G)+13 .3. γted(G) ≤ bp ∆(G)δ c.28Riyaz Ur Rehman A and A Mohamed Ismayil4. γted(G) ≥ d p1+∆(G)e.5. γted(G) ≤ dn+ ∆(G)−√2qe.The transversal eccentric dominating set, γted(G), upper transversal ec-centric dominating set and Γted(G) of standard graphs are tabulated.Graph FigureD - MinimumTED set.|D| = γted(G)γted(G)S - UpperTED set.|S| = Γted(G)Γted(G)Diamondgraph℘1℘4℘2 ℘3 {℘2, ℘3}. 2 {℘2, ℘3}. 2Tetrahedralgraph ℘2℘1℘3 ℘4{℘1, ℘2, ℘3, ℘4}. 4 {℘1, ℘2, ℘3, ℘4}. 4Clawgraph℘2 ℘3℘1℘4{℘1, ℘3},{℘2, ℘3},{℘3, ℘4}.2 {℘1, ℘2, ℘4}. 3(2,3)-Kinggraph℘2 ℘3℘1℘5℘4 ℘6{℘1, ℘2, ℘4},{℘1, ℘3, ℘4},{℘1, ℘3, ℘6},{℘1, ℘4, ℘5},{℘1, ℘4, ℘6},{℘2, ℘3, ℘6},{℘3, ℘4, ℘6},{℘3, ℘5, ℘6}.3{℘1, ℘2, ℘4},{℘1, ℘3, ℘4},{℘1, ℘3, ℘6},{℘1, ℘4, ℘5},{℘1, ℘4, ℘6},{℘2, ℘3, ℘6},{℘3, ℘4, ℘6},{℘3, ℘5, ℘6}.3Antennagraph℘2℘1℘3 ℘4℘5 ℘6{℘1, ℘2, ℘5},{℘1, ℘2, ℘6},{℘1, ℘3, ℘5},{℘1, ℘3, ℘6},{℘1, ℘4, ℘5},{℘1, ℘4, ℘6},{℘1, ℘5, ℘6},{℘2, ℘5, ℘6}.3 {℘1, ℘2, ℘3, ℘4}. 429Transversal eccentric domination in graphsGraph FigureD - MinimumTED set.|D| = γted(G)γted(G)S - UpperTED set.|S| = Γted(G)Γted(G)Pawgraph℘2℘3℘1℘4{℘1, ℘3},{℘2, ℘3}{℘3, ℘4}.2 {℘1, ℘2, ℘4}. 3Bullgraph℘3 ℘4℘5℘2℘1 {℘1, ℘2, ℘3},{℘1, ℘2, ℘4},{℘1, ℘2, ℘5},{℘1, ℘3, ℘4},{℘2, ℘3, ℘4}.3{℘1, ℘2, ℘3},{℘1, ℘2, ℘4},{℘1, ℘2, ℘5},{℘1, ℘3, ℘4},{℘2, ℘3, ℘4}.3Butterflygraph℘3℘2℘5℘1℘4{℘1, ℘2, ℘4},{℘1, ℘2, ℘5},{℘1, ℘3, ℘4},{℘1, ℘4, ℘5},{℘2, ℘3, ℘5},{℘2, ℘4, ℘5}.3{℘1, ℘2, ℘4},{℘1, ℘2, ℘5},{℘1, ℘3, ℘4},{℘1, ℘4, ℘5},{℘2, ℘3, ℘5},{℘2, ℘4, ℘5}.3Bannergraph℘3 ℘4℘1 ℘2℘5{℘2, ℘5}. 2 {℘2, ℘5}. 2Forkgraph℘2 ℘3℘1℘4 ℘5{℘1, ℘2, ℘5},{℘1, ℘4, ℘5},{℘2, ℘3, ℘5},{℘2, ℘4, ℘5}.3{℘1, ℘2, ℘3, ℘4},{℘1, ℘3, ℘4, ℘5}.4(3,2)-Tadpolegraph℘2 ℘3℘4℘1℘5{℘1, ℘4},{℘4, ℘5}.2 {℘1, ℘2, ℘3, ℘5}. 4Kite graph℘3℘4℘1℘5℘2 {℘2, ℘4}. 2 {℘1, ℘2, ℘3, ℘5}. 4(4,1)-Lollipopgraph℘3℘4℘1℘5℘2{℘1, ℘4},{℘2, ℘4},{℘3, ℘4},{℘4, ℘5}.2 {℘1, ℘2, ℘3, ℘5}. 430Riyaz Ur Rehman A and A Mohamed IsmayilGraph FigureD - MinimumTED set.|D| = γted(G)γted(G)S - UpperTED set.|S| = Γted(G)Γted(G)Housegraph ℘2 ℘3℘1℘4 ℘5{℘1, ℘2, ℘3},{℘1, ℘4, ℘5},{℘2, ℘3, ℘4},{℘2, ℘3, ℘5},{℘2, ℘4, ℘5},{℘3, ℘4, ℘5}.3{℘1, ℘2, ℘3},{℘1, ℘4, ℘5},{℘2, ℘3, ℘4},{℘2, ℘3, ℘5},{℘2, ℘4, ℘5},{℘3, ℘4, ℘5}.3House Xgraph ℘2 ℘3℘1℘4 ℘5{℘1, ℘2},{℘1, ℘3},{℘1, ℘4},{℘1, ℘5}.2 {℘2, ℘3, ℘4, ℘5}. 4Gemgraph℘1 ℘2℘5℘3 ℘4{℘1, ℘2, ℘3},{℘1, ℘2, ℘4},{℘1, ℘3, ℘4},{℘1, ℘3, ℘5}{℘2, ℘3, ℘4}{℘2, ℘4, ℘5}.3{℘1, ℘2, ℘3},{℘1, ℘2, ℘4},{℘1, ℘3, ℘4},{℘1, ℘3, ℘5},{℘2, ℘3, ℘4},{℘2, ℘4, ℘5}.3Dartgraph ℘3℘4℘1℘5℘2 {℘2, ℘3},{℘2, ℘4}.2{℘1, ℘2, ℘5},{℘1, ℘3, ℘4},{℘3, ℘4, ℘5}.3Cricketgraph ℘4 ℘5℘3℘1 ℘2{℘3, ℘4},{℘4, ℘5}.2{℘1, ℘2, ℘4},{℘1, ℘3, ℘5},{℘2, ℘3, ℘5}.3Pentatopegraph℘1℘4 ℘5℘2 ℘3 {℘1, ℘2, ℘3, ℘4, ℘5}. 5 {℘1, ℘2, ℘3, ℘4, ℘5}. 5Johnsonsolidskeleton 12graph℘2℘1℘3℘4 ℘5{℘1, ℘3}. 2{℘1, ℘2, ℘4, ℘5},{℘2, ℘3, ℘4, ℘5}.4Crossgraph℘3℘1℘2 ℘4℘5℘6{℘1, ℘3, ℘6},{℘2, ℘3, ℘6},{℘3, ℘4, ℘6},{℘3, ℘5, ℘6}.3{℘1, ℘2, ℘3, ℘4, ℘5},{℘1, ℘2, ℘4, ℘5, ℘6}.531Transversal eccentric domination in graphsGraph FigureD - MinimumTED set.|D| = γted(G)γted(G)S - UpperTED set.|S| = Γted(G)Γted(G)Netgraph ℘5℘6℘3 ℘4℘1 ℘2{℘1, ℘2, ℘5},{℘1, ℘2, ℘6},{℘1, ℘4, ℘6},{℘2, ℘3, ℘6}.3{℘1, ℘3, ℘4, ℘5},{℘2, ℘3, ℘4, ℘5},{℘3, ℘4, ℘5, ℘6}.4Fishgraph℘4℘2℘5℘1℘6℘3 {℘2, ℘3},{℘3, ℘5}.2 {℘1, ℘2, ℘4, ℘5, ℘6}. 5Agraph℘3 ℘4℘1℘5℘2℘6{℘1, ℘5, ℘6},{℘2, ℘5, ℘6}.3{℘1, ℘2, ℘3, ℘4, ℘5},{℘1, ℘2, ℘3, ℘4, ℘6}.5Rgraph℘3 ℘4℘1℘5℘2℘6{℘1, ℘2, ℘3},{℘2, ℘3, ℘4},{℘2, ℘3, ℘5},{℘2, ℘3, ℘6},{℘2, ℘5, ℘6}.3 {℘1, ℘3, ℘4, ℘5, ℘6}. 54-polynomialgraph℘2 ℘3℘1℘5℘4 ℘6{℘3, ℘4}. 2{℘1, ℘2, ℘3, ℘5, ℘6},{℘1, ℘2, ℘4, ℘5, ℘6}.5Octahedralgraph℘4℘3℘2℘1℘5 ℘6{℘1, ℘2, ℘3, ℘4},{℘1, ℘2, ℘3, ℘5},{℘1, ℘2, ℘3, ℘6},{℘1, ℘2, ℘4, ℘5},{℘1, ℘2, ℘5, ℘6},{℘1, ℘3, ℘4, ℘6},{℘1, ℘3, ℘5, ℘6},{℘1, ℘4, ℘5, ℘6},{℘2, ℘3, ℘4, ℘5},{℘2, ℘3, ℘4, ℘6},{℘2, ℘4, ℘5, ℘6},{℘3, ℘4, ℘5, ℘6}.4{℘1, ℘2, ℘3, ℘4},{℘1, ℘2, ℘3, ℘5},{℘1, ℘2, ℘3, ℘6},{℘1, ℘2, ℘4, ℘5},{℘1, ℘2, ℘5, ℘6},{℘1, ℘3, ℘4, ℘6},{℘1, ℘3, ℘5, ℘6},{℘1, ℘4, ℘5, ℘6},{℘2, ℘3, ℘4, ℘5},{℘2, ℘3, ℘4, ℘6},{℘2, ℘4, ℘5, ℘6},{℘3, ℘4, ℘5, ℘6}.43 ConclusionsIn this paper TED-set of a graph is defined. Theorems related to find the TED-number of different family of graphs are stated and proved. The upper and lower32Riyaz Ur Rehman A and A Mohamed IsmayilGraph FigureD - MinimumTED set.|D| = γted(G)γted(G)S - UpperTED set.|S| = Γted(G)Γted(G)3-prismgraph℘2℘3 ℘4℘1℘5 ℘6{℘1, ℘5, ℘6},{℘2, ℘3, ℘4}.3{℘1, ℘2, ℘3, ℘6},{℘1, ℘2, ℘4, ℘5},{℘1, ℘3, ℘4, ℘5},{℘1, ℘3, ℘4, ℘6},{℘2, ℘3, ℘5, ℘6},{℘2, ℘4, ℘5, ℘6}.4TED-number along with their respective sets of different standard graphs are tab-ulated. In future the comparative study of TED-set with eccentric dominating setwill be done. The properties of a TED-set related to graph operations such asunion, intersection, join and product of graphs will be explored.ReferencesA. Alwardi, S. Nayaka, et al. Transversal domination in graphs. Gulf Journal ofMathematics, 6(2), 2018.E. J. Cockayne and S. T. Hedetniemi. Towards a theory of domination in graphs.Networks, 7(3):247–261, 1977.F. Harary. Graph theory. Narosa Publishing House, New Delhi, 2001.A. M. Ismayil and A. R. U. Rehman. Accurate eccentric domination in graphs.Our Heritage, 68(4)(1, 2020):209–216, 2020a.A. M. Ismayil and A. R. U. Rehman. Equal eccentric domination in graphs.Malaya Journal of Matematik (MJM), 8(1, 2020):159–162, 2020b.T. Janakiraman, M. Bhanumathi, and S. Muthammai. Eccentric domination ingraphs. International Journal of Engineering Science, Computing and Bio-Technology, 1(2):1–16, 2010.O. Ore. Theory of graphs. Amer. Math. Soc. Colloq. Publ., 38 (Amer. Math. Soc.,Providence, RI), 38, 1996.33

Transversal Eccentric Domination in Graphs

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Transversal Eccentric Domination in Graphs

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