EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 2, 2024, 1082-1093 ISSN 1307-5543 – ejpam.com Published by New York Business Global Inverse Domination in X-Trees and Sibling Trees V. Shalini1, Indra Rajasingh1,∗ 1 Division of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Chennai, Tamilnadu, India Abstract. A set D of vertices in a graph G is a dominating set if every vertex not in D is adjacent to at least one vertex in D. The minimum cardinality of a dominating set in G is called the domination number and is denoted by γ(G). LetD be a minimum dominating set of G. If V −D contains a dominating set say D ′ of G, then D ′ is called an inverse dominating set with respect to D. The inverse domination number γ ′ (G) is the cardinality of a minimum inverse dominating set of G. A dominating set D is called a connected dominating set or an independent dominating set of G according as the induced subgraph ⟨D⟩ is connected or independent in G. The minimum of the cardinalities of the connected dominating sets of G or the independent dominating sets of G is called the connected domination number γc(G) or the independent domination number γi(G) respectively. In this paper, we determine the inverse domination numbers in X-Trees and Sibling Trees. We have also determined the independent domination numbers of both the trees and the connected domination number of Sibling Trees. A result on inverse domination number of some classes of Hypertrees is also included. 2020 Mathematics Subject Classifications: 05C69 Key Words and Phrases: Domination, Inverse domination, Connected Domination, Hypertrees, Sibling trees 1. Introduction Domination problems are studied to find sets of representatives to monitor communi- cation or electrical networks and in land surveying where it is necessary to minimize the number of places a surveyor must stand to take height measurements for an entire region [24]. It also plays a vital role in parallel processing and supercomputing, which continue to exert great influence on the development of modern science and engineering. In any net- work, dominating sets are central sets and hence they play a key role in routing problems associated with parallel computing [22]. A non-empty subset D ⊆ V (G) is a dominating set if each vertex in V (G) − D is adjacent to at least one vertex in D. Such a set with minimum cardinality yields the domination number of a graph G and is denoted by γ(G) ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i2.5038 Email addresses: shalu.maths@gmail.com (V. Shalini), indrarajasingh@yahoo.com (I. Rajasingh) https://www.ejpam.com 1082 © 2024 EJPAM All rights reserved. V. Shalini, I. Rajasingh / Eur. J. Pure Appl. Math, 17 (2) (2024), 1082-1093 1083 [13]. A dominating set D is said to be a connected dominating set or an independent dom- inating set if the induced subgraph ⟨D⟩ is connected or independent in G. The minimum of the cardinalities of the connected dominating sets of G or independent dominating sets of G is called the connected domination number of G denoted by γc(G) or the indepen- dent domination number denoted by γi(G)[18]. The study of connected domination has extensive application in the study of routing problems and virtual backbone based routing in wireless networks[4, 15, 25]. Determining if an arbitrary graph has a dominating set of a given size is a well-known NP -complete problem [10]. Finding optimum dominating sets in networks has always been challenging. Let D be a minimum dominating set of G. If V − D contains a dominating set say D ′ of G, then D ′ is called an inverse dominating set with respect to D. The inverse dom- ination number γ ′ (G) is the order of a smallest inverse dominating set in G [14]. Inverse domination in graphs introduced by Kulli and Sigarkanti [14] in 1991 plays a major role in reliable communication and electrical networks. Suppose D is a minimum dominating set in a graph G and some nodes of D fail, the inverse dominating set plays the role of D. Domke, Dunbar, and Markus (Ars Combin. 72 (2004), 149–160)[6] conjectured that the inverse domination number of G is at most the independence number of G. The above conjecture has been proved for special families of graphs, including claw-free graphs, bi- partite graphs, split graphs, very well covered graphs, chordal graphs and cactus graphs in[8]. There are some graphs with equal domination and inverse domination numbers are identified by T.TamizhChelvam [2]. Inverse domination number of circulant graph proved by V.Cynthiya in [3]. Also we identified domination and inverse domination numbers for Wrapped butterfly network, Lollipop graph, Fly graph and Jellyfish graph in [19–21]. In this paper, we determine the inverse domination, independent domination and con- nected domination numbers in sibling tree networks and also find the domination, inde- pendent domination and inverse domination numbers for the X-tree networks. The inverse domination number of some classes of hypertrees is included. 2. Domination and Inverse Domination in X-Trees Efficient inter-processor communication is one of the crucial issues in multiprocessor systems [1, 7, 9, 12, 23]. Multiple processors are interconnected in a tightly coupled, hier- archical, tree-structured network. An X-tree [5] is a complete binary tree with additional edges to connect consecutive nodes on the same level of the tree so that the vertices on each level induce a path. Edges on such paths are called horizontal edges. Horizontal edges are of two types: sibling edges and cousin edges. A sibling edge denotes a horizontal edge that connects two vertices with the same parent and a cousin edge denotes any of the remaining horizontal edges. Two sibling edges are said to be adjacent if there is exactly one cousin edge between them. The tree edges are addressed as vertical edges. The struc- tural characteristic of X-tree was identified in [5]. The root of X(k) is considered to be at Level 0. Vertices at level k are called leaf vertices. The vertices of X(k) other than the root and the leaf vertices are called internal vertices. A k-level X-tree or a 2k-leaf X-tree will be denoted by X(k). A k-level X-tree has 2k+1 − 1 vertices and 2k+2 − k − 4 edges V. Shalini, I. Rajasingh / Eur. J. Pure Appl. Math, 17 (2) (2024), 1082-1093 1084 Figure 1: (a) X(3) with labels. (b) Subgraph H in Lemma 2. [16]. X-trees are fault tolerant variants of the basic tree network and have been the focus of more recent implementation in massively parallel systems. In this section, we determine the domination number of X(k), k ≥ 1. For convenience, we label the vertices of X(3) as in Figure 1(a). Lemma 1. The domination number of X(3) is given by γ(X(3)) = 4. Proof. Let D be a dominating set of X(3). Three distinct vertices are necessary in D to dominate the degree 2 vertices u, l3 and r3. Refer Figure 1(a). To optimize the cardinality of the neighbourhoods of vertices adjacent to u, l3 and r2, we choose l1, r2 (or 8) and l2 (or 3) in D. Another possibility is r1, l2 (or 3) and r2 (or 8). In either case, the number of dominated vertices is 7. To dominate the subgraph induced by the remaining 5 vertices, we require at least one more vertex in D. Thus |D|≥ 4. It is easy to verify that {l1, r2, 3, 6} is a dominating set of X(3). Hence γ(X(3)) = 4. Remark 1. {r1, l2, 5, 8} is also a minimum dominating set of X(3). Lemma 2. A minimal dominating set of X(3) that contains the root of X(3) is of cardi- nality 5. Proof. Let D be a dominating set of X(3) that contains u. Then u dominates l1 and r1. Consider the subgraph H induced by Level 2 and Level 3 vertices of X(3) which are yet to be dominated. See Figure 1(b). To dominate the degree 2 vertices l3 and r3, it is necessary to include 2 vertices of H in D. The possibilities that do not include l3 or r3 are {l2, r2}, {l2, 8}, {r2, 3} and {3, 8}. In all cases, the remaining vertices to be dominated induce a path of length 3. As there are two pendant vertices, two more vertices from the path are to be included in D. On the otherhand, the possibilities that include l3 or r3 are {l3, r3}, {l3, r2}, and {l2, r3}. In all these cases, the remaining vertices to be dominated are 1,2,4,5,6 and 7. The subgraph induced by these vertices require 2 vertices to be included in D. Thus |D|= 5. V. Shalini, I. Rajasingh / Eur. J. Pure Appl. Math, 17 (2) (2024), 1082-1093 1085 Figure 2: (a) Subgraph H1 dominated by circled vertices. (b)Subgraph H2 dominated by circled vertices. Lemma 3. A minimal dominating set of X(3) containing (i) {l1, l2}, (ii){l2, l3}, or (iii) {l3, l1} is of cardinality 5. Proof. Let D be a minimal dominating set of X(3). Suppose D contains {l1, l2} or {l3, l1}. Now if N [S] denotes the closed neighbourhood of a set S of vertices, then N [{l1, l2}] = {u, l1, l2, l3, r1, 1, 3} = N [{l3, l1}]. The subgraphH1 induced by the remaining vertices require 3 vertices to be included in D to dominate all vertices of the subgraph. See Figure 2(a). Again, suppose D contains {l2, l3}. We have N [{l2, l3}] = {l1, l2, l3, 1, 3}. The subgraph H2 induced by the remaining vertices require 3 vertices to be included in D to dominate all vertices of the subgraph. See Figure 2(b). In either cases, we have |D|= 5. We now proceed to determine the domination number of X(k), k ≥ 3. Let H be the subgraph induced by the vertices in Levels k, k − 1, k − 2 and k − 3 of X(k), k ≥ 3. H has the following properties: (i) V (H) = ⋃2k−3 i=1 Vi such that the subgraph induced by Vi is isomorphic to X(3), 1 ≤ i ≤ 2k−3. Let these copies of X(3) be named H1, H2,..., H2k−3 from left to right as shown in Figure 3. Let the roots of Hi, 1 ≤ i ≤ 2k−3 be labeled u1, u2,..., u2k−3 respectively. Figure 3: Subgraph induced by vertices in Levels 2,3,4,5 of X(5) Let the leftmost descendants of ui be labeled li1, li2 and li3; similarly let the rightmost descendants of ui be labeled ri1, ri2 and ri3, 1 ≤ i ≤ 2k−3. See Figure 3. Let the 2 unlabeled vertices in Level k− 1 in each Hi be labeled as (i− 1)8+ 1 and (i− 1)8+ 2 and the 6 unlabeled vertices in Level k of each Hi as (i− 1)8 + 3, (i− 1)8 + 4,..., (i− 1)8 + 8 V. Shalini, I. Rajasingh / Eur. J. Pure Appl. Math, 17 (2) (2024), 1082-1093 1086 Figure 4: (a) Subgraph H1 in Case 1. (b) Subgraph H2 in Case 1.(c) Subgraph H2 in Case 2. from left to right, 1 ≤ i ≤ 2k−3. Lemma 4. Let H be the subgraph of X(k) induced by Levels k, k − 1, k − 2 and k − 3 of X(k), k ≥ 3. Then γ(H) = 2k−1. Proof. Due to symmetricity and the fact that vertices of Hi−1 and Hi+1 can dominate vertices in Hi, it is enough to consider the domination parameter in the subgroup H1 ∪ H2 ∪H3. Case 1. Consider H1 ∪H2. Suppose u1, r11, r12 and r13 are dominated by vertices that are not in H1. Then the subgraph of H1 induced by the remaining vertices of H1 require 3 vertices of H1 in any dominating set D of H. See Figure 4(a). In this case necessarily {u2, l21, l22, l23} ⊆ D. To dominate the remaining vertices in H2, 3 more vertices in H2 are to be included in D. See Figure 4(b). Thus to dominate H1 ∪H2 at least 9 vertices of H1 ∪H2 are to be included in D. Case 2: Consider H1 ∪ H2 ∪ H3. Suppose u2, l21, l22, l23, r21, r22 and r23 are already dominated by vertices that are not in H2. Then the subgraph of H2 induced by the remaining vertices of H2 require 2 vertices of H2 in D. See Figure 4(c). In this case necessarily {r11, r12, r13, l31, l32, l33} ⊆ D. Three more vertices in each of H1 and H3 are to be included in D to dominate all the vertices in H1 ∪ H2 ∪ H3. Thus to dominate H1 ∪H2 ∪H3 at least 6 + 2 + 6 = 14 vertices of H1 ∪H2 ∪H3 are to be included in D. By virtue of Lemmas 2 and 3 and arguments similar to the above cases, we claim that selecting 4 vertices in each Hi, 1 ≤ i ≤ 2k−3 as in Lemma 1 yields a minimum dominating set D of X(k). By Lemma 1, D is a dominating set of H. To prove that D is a minimum dominating set, we need to consider only H1 ∪ H2 ∪ H3 and consider the dominating sets D1 = {l11, r12, 3, 6}, D′ 1 = {r11, l12, 5, 8} of H1 and D3 = {l31, r32, 19, 22}, D′ 3 = {r31, l32, 21, 24} of H3. If D1 and D′ 3 are in D, then we observe that r12 dominates l22 and l32 dominates r22. The subgraph of H2 induced by the remaining vertices requires more than 4 vertices to dominate the subgraph. See Figure 5(a). On the other hand if D1 and D3 are in D then r12 dominates l22 and l31 dominates r21. In this case, the subgraph of H2 induced by the remaining vertices require 4 vertices to dominate the subgraph . See Figure 5(b). In either case, the number of vertices dominating H2, considering the already dominated vertices of H2, is not less than 4. Thus 4 vertices from each Hi, 1 ≤ i ≤ 2k−3 is a minimum count in D. Hence D is a minimum dominating set and |D|= 4× 2k−3 = 2k−1. V. Shalini, I. Rajasingh / Eur. J. Pure Appl. Math, 17 (2) (2024), 1082-1093 1087 Figure 5: (a) l22 and r22 of H2 are already dominated. (b) l22 and r21 of H2 are already dominated. Theorem 1. Let G be the X-Tree of dimension k and it is denoted by X(k), k ≥ 0, Then γ(X(k)) =  2k+3−4 15 ; k + 1 ≡ 0 (mod 4) 2k+3+7 15 ; k + 1 ≡ 1 (mod 4) 2k+3−1 15 ; k + 1 ≡ 2 (mod 4) 2k+3−2 15 ; k + 1 ≡ 3 (mod 4) Proof. By Lemma 2 and Lemma 4, it is clear that a count of 4 levels starting from the last Level k of X(k) contributes 2k−1 vertices to any minimum dominating set D of X(k). Deleting these 4 levels from X(k) yields X(k − 4). Applying Lemma 4 and delet- ing the last 4 levels repeatedly we are left with X(0), X(1), X(2) or X(3) according as k + 1 ≡ 0, 1, 2 or 3 (mod 4) respectively. We have γ(X(0)) = γ(X(1)) = 1, γ(X(2)) = 2. Thus we compute the domination number of X(k) as follows. when k + 1 ≡ 0 (mod 4), |D|= 2k−1 + 2k−5 + ...+ 210 + 26 + 22 = 2k+3−4 15 when k + 1 ≡ 1 (mod 4), |D|= (2k−1 + 2k−5 + ...+ 27 + 25) + 1 = 2k+3+7 15 when k + 1 ≡ 2 (mod 4), |D|= (2k−1 + 2k−5 + ...+ 28 + 24) + 1 = 2k+3−1 15 and when k + 1 ≡ 3 (mod 4), |D|= (2k−1 + 2k−5 + ...+ 211 + 27) + 2 = 2k+3−2 15 Notations: The sets of vertices {l1, r2, 3, 6} and {r1, l2, 5, 8} in a copy of X(3) as in Figure 1(a) are disjoint dominating sets of X(3). We refer to them as D-Twin sets. Theorem 2. Let X(k) be the X-tree of dimension k ≥ 0. Then γ′(X(k)) = γ(X(k)), k ≥ 0. Proof. We construct two minimum dominating sets D and D′ of X(k), k ≥ 0 as follows: Include one set of D-Twin vertices in D and the other set of D-Twin vertices in D′ from each copy of X(3) considered in Theorem 1. The vertices that are not covered by these copies of X(3) induce X(0), X(1) or X(2) according as k + 1 ≡ 1, 2 or 3(mod 4). See Figure 6. When k + 1 ≡ 1 or 2(mod 4), no D-Twin set includes l or r. Hence put l in D and r in D′. On the otherhand, when k + 1 ≡ 3(mod 4), none of the Level 2 vertices in V. Shalini, I. Rajasingh / Eur. J. Pure Appl. Math, 17 (2) (2024), 1082-1093 1088 Figure 6: Subgraph induced by leftout vertices when k + 1 ≡ 1, 2, 3 or (mod 4) Figure 7: (a) Levels of ST (3). (b) ST (2) with labels. X(2) is included in any D-Twin set. Hence include {l1, r2} in D and {r1, l2} in D′. Thus D and D′ are dominating sets of the same cardinality as the one constructed in Theorem 1. Thus γ′(X(k)) = γ(X(k)), k ≥ 0. We note that the dominating set D of X(k), k ≥ 0 constructed in Theorem 1 is an independent dominating set. Thus we have the following result. Theorem 3. Let X(k) be the X-Tree of dimension k. Then γ(X(k)) = γ′(X(k)) = γi(X(k)). 3. Inverse and Connected Domination in Sibling Trees A Sibling tree ST (r) of dimension r is obtained from the complete binary tree T (r) of height r by adding edges called sibling edges joining left and right children of the same parent node. The root node is at Level 0. Level i vertices are the children of vertices in Level i− 1, 1 ≤ i ≤ r. ST (r) has 2r+1 − 1 vertices and 3(2r − 1) edges, See Figure 7(a). Notation: Let H be a graph, isomorphic to ST (2) as shown in the Figure 7(b). We call the pair of vertices u and v as irregular D-Twins as u and v together dominate all the vertices of H. Similarly, x and y form another pair of irregular D-Twins. Incidentally, the vertices u and x form a pair of regular of D-Twins. We refer to vertex w as the apex vertex of H. Remark 2. The domination number of ST (r), r ≥ 0, has been determined in [17] making use of regular D-Twins. In this section, we make use of irregular D-Twins, leading to the computation of inverse domination number of ST (r). V. Shalini, I. Rajasingh / Eur. J. Pure Appl. Math, 17 (2) (2024), 1082-1093 1089 Lemma 5. Let H be an induced subgraph from the last three levels of ST (r), r ≥ 2, isomorphic to ST (2). Then γ(H) = 2. Proof. Even if the apex vertex w of H is already dominated by some vertex from V (ST (r)) − V (H), there is no vertex of degree 5 in H that dominates the remaining vertices. Thus γ(H) ≥ 2. Further, any pair of irregular D-Twin vertices in H dominates all vertices in H. Hence γ(H) = 2. Theorem 4. Let G be the sibling tree ST (r) of dimension r ≥ 0. Then γ(ST (r)) =  1 7(2 r+2 + 3) ; r ≡ 0 (mod 3) 1 7(2 r+2 − 1) ; r ≡ 1 (mod 3) 1 7(2 r+2 − 2) ; r ≡ 2 (mod 3) Proof. We partition the levels L0, L1, L2, L3,..., Lr of ST (r) into maximum number of disjoint 3-levels, beginning from Lr. Clearly the left out levels will be L0 when r (mod 3) = 0; L0 and L1 when r (mod 3) = 1. In otherwords, we have the partitioning set P of the levels of ST (r) as follows: (i) When r (mod 3) = 0, P = {L0} ∪ ⋃⌊ r 3 ⌋−1 i=0 {Lr−3i−2, Lr−3i−1, Lr−3i}; (ii) When r (mod 3) = 1, P = {L0, L1} ∪ ⋃⌊ r 3 ⌋−1 i=0 {Lr−3i−2, Lr−3i−1, Lr−3i}; (iii) When r (mod 3) = 2, P = ⋃⌊ r 3 ⌋ i=0{Lr−3i−2, Lr−3i−1, Lr−3i}. We note that Lr−3i−2 has 2r−3i−2 vertices for any i, 0 ≤ i ≤ ⌊ r3⌋ − 1. Hence Lr−3i−2, Lr−3i−1, Lr−3i together induce 2r−3i−2 disjoint copies of ST (2), 0 ≤ i ≤ ⌊ r3⌋ − 1. Thus the number α of vertex disjoint copies of ST (2) in ST (r), r ≥ 3, is α = ⌊ r 3 ⌋−1∑ i=0 2r−3i−2 = 2r+1 7 (23⌊ r 3 ⌋ − 1 2⌊ r 3 ⌋ ) = 1 7 (2r+1 − 2r(mod 3)+1) By Lemma 5, each ST (2) contributes 2 vertices to any minimum dominating set D of ST (r). Hence γ(ST (r)) = 2α + 1 for r ≡ 0, 1(mod 3) and γ(ST (r)) = 2α + 2 for r ≡ 2(mod 3). Hence the result. Theorem 5. Let G be the sibling tree ST (r) of dimension r ≥ 0. Then γ′(ST (r)) = γ(ST (r)), r ≥ 0. Proof. We construct two minimum dominating sets D and D′ of ST (r), r ≥ 0 as follows. Include one pair of irregular D-Twin vertices in D and the other pair of D-Twin vertices in D′ from each of the copies of ST (2) considered in Theorem 4. The vertices of ST (r) that are not covered by these copies of ST (2) induce ST (0) or ST (1) according as r ≡ 0 or 1 mod 3. Since the children u and x of the root node w (Refer Figure 7(b)) of ST (r) do not belong to any D-Twin set include u in D and x in D′. Thus D and D′ are dominating sets of the same cardinality as the one constructed in Theorem 4. Thus γ ′ (ST (r)) = γ(ST (r)), r ≥ 0. V. Shalini, I. Rajasingh / Eur. J. Pure Appl. Math, 17 (2) (2024), 1082-1093 1090 We note that the dominating set of ST (r) constructed in Theorem 4 is an independent dominating set. Thus we have the following result. Theorem 6. Let G be the Sibling tree ST (r) of dimension r ≥ 0. Then γ(ST (r)) = γ ′ (ST (r)) = γi(ST (r)). Theorem 7. Let G be the Sibling tree ST (r), r ≥ 2, then γc(G) = 2r − 2. Proof. Let D be a connected dominating set of ST (r), r ≥ 2. Let v be an arbitrary vertex of ST (r) in Level i, 1 ≤ i ≤ r − 1. Then v is a cut vertex of ST (r). Let Gv and G′ v be the components of ST (r) \ {v}. Suppose v /∈ D. Let D(Gv) = V (Gv) ∩ D and D(G′ v) = V (G′ v)∩D. ThenD = D(Gv)∪D(G′ v). Choose x ∈ D(Gv) and y ∈ D(G′ v). Since D is connected, there exists a path between x and y in D. This path has to necessarily pass through v. But v /∈ D and hence there is no path between x and y in D, a contradiction to the connectedness of D. This implies v ∈ D. Thus any internal vertex v in Level i, 1 ≤ i ≤ r − 1 is a member of D. Obviously, these vertices also dominate G. Hence γc(G) = (2r+1 − 1)− 1− 2r = 2r − 2. 4. Inverse Domination in Hypertree Networks The fundamental skeleton of a hypertree is a complete binary tree Tn of height n. Here the nodes of the tree are numbered as follows: The root node has label 1. The root is supposed to be at level 0. Labels of left and right children are formed by appending 0 and 1, respectively to the labels of the parent node. The decimal and binary labels of the hypertree are given in Figure 8. Here the children of the node x are labelled as 2x and 2x+1. Additional links in a hypertree are horizontal and two nodes are joined in the same level i of the tree if their label difference is 2i−1. We denote an n-level hypertree as HT (n). It has 2n+1− 1 vertices and 3(2n− 1)edges. Hypertree is a multiprocessor interconnection topology which has a frequent data exchange in algorithms such as sorting and Fast and Fourier Transforms (FFT’s) [11]. Figure 8: HT (3) with decimal and binary labels. Following the lines of Theorem 4, we have obtained γ′(HT (n)) for n ≡ 0, 1(mod 3). An independent proof has been given in [17] to obtain γ(HT (n)). V. Shalini, I. Rajasingh / Eur. J. Pure Appl. Math, 17 (2) (2024), 1082-1093 1091 Theorem 8. Let G be the hypertree HT (r) of dimension r ≥ 0. Then γ′(HT (r)) = γ(HT (r)) = { 1 7(2 r+2 + 3); r ≡ 0 (mod 3) 1 7(2 r+2 − 1); r ≡ 1 (mod 3) Conjecture 8.1. Let G be the hypertree HT (r), r ≥ 0. Then γ′(HT (r)) = γ(HT (r)) + 1 = 1 7(2 r+2 − 2) + 1, for r ≡ 2 (mod 3). Figure 9: Plotting of Domination numbers. 5. Conclusion In this paper, we have obtained domination, inverse domination and independent dom- ination numbers of the X-Tree Networks. Similarly, we have obtained the domination, in- verse domination, independent domination and connected domination numbers of Sibling Tree Networks. We have also obtained the inverse domination number of a few classes of Hypertree networks. 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