Ratio Mathematica Volume 45, 2023 Inverse Domination Parameters of Jump Graph S. Santha1 G.T. Krishna Veni2 Abstract Let 𝐺 = (𝑉, 𝐸) be a connected graph. Let 𝐷 be a minimum dominating set in 𝐺. If 𝑉 βˆ’ 𝐷 contains a dominating set 𝐷′ of 𝐺, then 𝐷′ is called an inverse dominating set with respect to 𝐷. Theminimum cardinality of an inverse dominating set of 𝐺 is called inverse domination number of 𝐺. In this article, we determine inverse domination parameters of jump graph of a graph. Keywords: domination number, inverse domination number, non-split inverse domination number, connected inverse domination number, jump graph. 2010 AMS subject classification: 05C693. 1Assistant Professor, Department of Mathematics, Government Arts and Science, Konam, Nagercoil- 629004, Kanyakumari Dt, Tamil Nadu, India. Email: santhawilliam14@gmail.com. 2 Register Number: 18221172092026, Research Scholar, PG & Research Dept. of Mathematics, Rani Anna Govt. College for Women, Tirunelveli-8, Affiliated to Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli-627012, Tamil Nadu, India. Email: krishnavenisiva55@gmail.com. 3Received on July 10, 2022. Accepted on October 15, 2022. Published on January 30, 2023. doi: 10.23755/rm. v45i0.1001. ISSN: 1592-7415. eISSN: 2282-8214. Β©The Authors. This paper is published under the CC-BY license agreement. 135 S. Santha and G.T. Krishna Veni 1.Introduction The undirected graph𝐺 = (𝑉, 𝐸) discussed in this paper is simple and connected. The order and size are denoted by 𝑛 and π‘š respectively. For basic graph theoretical reference, we refer [2]. If 𝑒𝑣 is an edge of 𝐺, then two vertices 𝑒 and 𝑣 are said to be adjacent. If 𝑒𝑣 ∈ 𝐸(𝐺), then 𝑒 is 𝑣′𝑠 neighbour, and the set of 𝑣′𝑠 neighbours is denoted by 𝑁(𝑣). Vertex 𝑣 ∈ 𝑉 has a degree of 𝑑𝑒𝑔(𝑣) = |𝑁(𝑣)|. If 𝑑𝑒𝑔(𝑣) = 𝑛 βˆ’ 1, a vertex v is referred to as a universal vertex. Globally, the domination conception in graphs start off its root in 1850s along the concern of certain chess players. Domination has diverse function consists of the morphological analysis, social network theory, CCTV installation and the most generally argued is the computer network communication. The aforementioned network comprises communication links among a firm number of slots. The trouble is to prime a least number of location where transmitters are installed such that a network as a whole united by a direct communication link to the transmitter site. In alternative words, the issue is to locate the least dominant set in the graph that corresponds to this network. V.R. Kulli and C. Sigarkanthi [10] were the first to propose the idea of an inverse domination number. A set 𝐷 of vertices in a graph 𝐺 = (𝑉, 𝐸) is a dominating set if every vertex in 𝑉 βˆ’ 𝐷is adjacent to some vertex in 𝐷. The domination number 𝛾(𝐺)of 𝐺is the minimum cardinality of a dominating set of𝐺. If 𝑉 βˆ’ 𝐷 contains a dominating set 𝐷′ of 𝐺, then 𝐷′ is called an inverse dominating set with respect to 𝐷. The inverse domination number 𝛾 β€²(𝐺)of 𝐺is the minimum cardinality of an inverse dominating set of 𝐺. The dominating set 𝑆 of 𝐺 is connected dominating set of 𝐺 if induced sub graph βŒ©π‘†βŒͺ is connected. The connected domination number 𝛾𝑐 (𝐺) of 𝐺 is referred to as a minimum cardinality of connected dominating set. The dominating set 𝑆 of 𝐺 is a non-split dominating set of 𝐺 if induced sub graph βŒ©π‘‰ βˆ’ 𝑆βŒͺ is connected. The non-split domination number 𝛾𝑛𝑠(𝐺) of 𝐺 is referred to as a minimum cardinality of non-split dominating set.[7] The n-sunlet graph is a graph on 2𝑛 vertices isobtained by attaching 𝑛-pendant edges to the cycle 𝐢𝑛 and it is denoted by 𝑆𝑛. Let 𝑃𝑛 be a path graph in 𝑛 vertices. The comb graph is defined as 𝑃𝑛 ʘ𝐾1.It has 2𝑛 vertices and 2𝑛 βˆ’ 1 edges. Fan graph 𝐹𝑛n β‰₯ 2 determined by joining all vertices of a path 𝑃𝑛 to a different vertex, called centre. Thus 𝐹𝑛has 𝑛 + 1 vertices, such as 𝑒, 𝑒1, 𝑒2, 𝑒3, … . 𝑒𝑛 and (2𝑛 βˆ’ 1) edges, such as 𝑒𝑒𝑖 , 1 ≀ 𝑖 ≀ 𝑛 – 1. [12] The line graph 𝐿(𝐺) of 𝐺 has the edges of 𝐺 as its vertices which are adjacent in 𝐿(𝐺) if and only if the corresponding edges are adjacent in 𝐺. We call the complement of line graph 𝐿(𝐺)as the jump graph 𝐽(𝐺) of 𝐺, found in [11]. The jump graph 𝐽(𝐺) of a graph 𝐺 is the graph defined on𝐸(𝐺) and in which two vertices are adjacent if and only if they are not adjacent in 𝐺. Since both 𝐿(𝐺) and 𝐽(𝐺) are defined on the edge set of a graph 𝐺. 136 Inverse Domination Parameters of Jump 2. Main Results Inverse domination number of jump graph of 𝒏–sunlet graph Theorem 2.1. For the graph 𝐺 = 𝐽(𝑆𝑛) (𝑛 β‰₯ 4), π›ΎκžŒ (𝐺)) = 2. Proof. For 𝑛 β‰₯ 4, the number of vertices of 𝑛-sunlet graph is 2𝑛. Then it has 2𝑛 edges. Let 𝐺 = 𝐽(𝑆𝑛). The number of vertices of jump graph of the 𝑛-sunlet graph is 2𝑛. Let the vertices of the graph is labeled as {𝑣1, 𝑣2, … , 𝑣𝑛 , 𝑒1, 𝑒2, 𝑒3, … . 𝑒𝑛}, Since 𝐺 contains no universal vertices, 𝛾(𝐺) β‰₯ 2. Let 𝐷 = {𝑒1, 𝑒3}. Then 𝐷 is a dominating set of 𝐺 and so 𝛾(𝐺)) = 2. Let 𝐷′ = {𝑒2, 𝑒4}. Then 𝐷 β€² is a inverse dominating set of 𝐺 so that 𝛾 β€²(𝐺) = 2. ∎ Theorem 2.2. For the graph 𝐺 = 𝐽(𝑆𝑛) (𝑛 β‰₯ 4), 𝛾𝑛𝑠 β€² (𝐺)) = 2. Proof. For 𝑛 β‰₯ 4, the number of vertices of 𝑛-sunlet graph is 2𝑛. Then it has 2𝑛 edges. Let 𝐺 = 𝐽(𝑆𝑛). The number of vertices of jump graph of the 𝑛-sunlet graph is 2𝑛. Let the vertices of the graph is labeled as {𝑣1, 𝑣2, … , 𝑣𝑛 , 𝑒1, 𝑒2, 𝑒3, … . 𝑒𝑛}, Since 𝐺 contains no universal vertices, 𝛾(𝐺) β‰₯ 2. Let 𝐷 = {𝑒1, 𝑒3}. Then 𝐷 is a dominating set of 𝐺 and so 𝛾(𝐺) = 2. Let 𝐷′ = {𝑒2, 𝑒4}. Then 𝐷 β€² is a non-split inverse dominating set of 𝐺 so that 𝛾𝑛𝑠 β€² (𝐺) = 2. ∎ Theorem 2.3. For the graph 𝐺 = 𝐽(𝑆𝑛) (𝑛 β‰₯ 4), 𝛾𝑐 β€² (𝐺) = 2. Proof. For 𝑛 β‰₯ 4, the number of vertices of 𝑛-sunlet graph is 2𝑛. Then it has 2𝑛 edges. Let 𝐺 = 𝐽(𝑆𝑛). The number of vertices of jump graph of the 𝑛-sunlet graph is 2𝑛. Let the vertices of the graph is labeled as {𝑣1, 𝑣2, … , 𝑣𝑛 , 𝑒1, 𝑒2, 𝑒3, … . 𝑒𝑛}, Since 𝐺 contains no universal vertices, 𝛾(𝐺) β‰₯ 2. Let 𝐷 = {𝑒1, 𝑒3}. Then 𝐷 is a dominating set of 𝐺 and so 𝛾(𝐺)) = 2. Let 𝐷′ = {𝑒2, 𝑒4}. Then 𝐷 β€² is a connected inverse dominating set of 𝐺 so that 𝛾𝑐 β€²(𝐺) = 2. ∎ Inverse domination number of jump graph of comb graph Theorem 2.4. For the graph 𝐺 = 𝐽(𝑃𝑛 βŠ™ 𝐾1) (𝑛 β‰₯ 4), 𝛾 β€²(𝐺) = 2. Proof. For 𝑛 β‰₯ 4, the number of vertices of combgraph is 2𝑛. Then it has 2𝑛 βˆ’ 1 edges. Let 𝐺 = 𝐽(𝑃𝑛 βŠ™ 𝐾1). The number of vertices of jump graph of the comb graph is 2𝑛 βˆ’ 1. Let the vertices of the graph is labeled as {𝑣1, 𝑣2, … , π‘£π‘›βˆ’1, 𝑒1, 𝑒2, 𝑒3, … . 𝑒𝑛}, Since 𝐺 contains no universal vertices, 𝛾(𝐺) β‰₯ 2. Let 𝐷 = {𝑒1, 𝑣1}. Then 𝐷 is a dominating set of 𝐺 and so 𝛾(𝐺) = 2. Let 𝐷′ = {𝑒2, 𝑒4}. Then 𝐷 β€² is a inverse dominating set of 𝐺 so that 𝛾 β€²(𝐺) = 2. ∎ Theorem 2.5. For the graph 𝐺 = 𝐽(𝑃𝑛 βŠ™ 𝐾1) (𝑛 β‰₯ 4), 𝛾𝑛𝑠 β€² (𝐺) = 2. Proof. For 𝑛 β‰₯ 4, the number of vertices of comb graph is 2𝑛. Then it has 2𝑛 βˆ’ 1 edges. Let 𝐺 = 𝐽(𝑃𝑛 βŠ™ 𝐾1). The number of vertices of jump graph of the comb graph is 2𝑛 βˆ’ 1. Let the vertices of the graph is labeled as {𝑣1, 𝑣2, … , π‘£π‘›βˆ’1, 𝑒1, 𝑒2, … , 𝑒𝑛}. Since 𝐺 contains no universal vertices, 𝛾(𝐺) β‰₯ 2. Let 𝐷 = {𝑒1, 𝑣1}. Then 𝐷 is a dominating set of 𝐺 and so 𝛾(𝐺) = 2. Let 𝐷′ = {𝑒2, 𝑒4}. Then 𝐷 β€² is a non-split inverse dominating 137 S. Santha and G.T. Krishna Veni set of 𝐺 so that 𝛾𝑛𝑠 β€² (𝐺) = 2. ∎ Theorem 2.6. For the graph 𝐺 = 𝐽(𝑃𝑛 βŠ™ 𝐾1) (𝑛 β‰₯ 4), 𝛾𝑐 β€²(𝐺) = 2. Proof. For 𝑛 β‰₯ 4, the number of vertices of comb graph is 2𝑛. Then it has 2𝑛 βˆ’ 1 edges. Let 𝐺 = 𝐽(𝑃𝑛 βŠ™ 𝐾1). The number of vertices of jump graph of the comb graph is 2𝑛 βˆ’ 1. Let the vertices of the graph is labeled as {𝑣1, 𝑣2, … , π‘£π‘›βˆ’1, 𝑒1, 𝑒2, … , 𝑒𝑛}, Since 𝐺 contains no universal vertices, 𝛾(𝐺)) β‰₯ 2. Let 𝐷 = {𝑒1, 𝑣1}. Then 𝐷 is a dominating set of 𝐺 and so 𝛾(𝐺) = 2. Let 𝐷′ = {𝑒2, 𝑒4}. Then 𝐷 β€² is a connected inverse dominating set of 𝐺 so that 𝛾𝑐 β€²(𝐺) = 2. ∎ Inverse domination number of jump graph of fan graph Theorem 2.7. For fan graph 𝐺 = 𝐽(𝐹𝑛) (𝑛 β‰₯ 5), 𝛾 β€²(𝐺) = { 3 𝑖𝑓 𝑛 = 3,4 2 𝑖𝑓 𝑛 β‰₯ 5 . Proof. The number of vertices of fan graph is 𝑛 + 1. Then it has 2𝑛 βˆ’ 1 edges. Let 𝐺 = 𝐽(𝐹𝑛 ). The number of vertices of jump graph of the fan graph is 2𝑛 βˆ’ 1. Let the vertices of the graph is labeled as {𝑒1, 𝑒2, … , 𝑒𝑛, 𝑣1, 𝑣2, … , π‘£π‘›βˆ’1}, Since 𝐺 contains no universal vertices, 𝛾(𝐺)) β‰₯ 2. Let 𝑛 = 3. It is easily verified that no two element subsets of𝐽(𝐹𝑛 ) is not a 𝛾-set of 𝐺 and so 𝛾(𝐺) β‰₯ 3.Let 𝐷 = {𝑣1, 𝑣3, 𝑣4}. Then 𝐷 is a 𝛾-set of 𝐺 so that 𝛾(𝐺)) = 3. Let 𝐷′ = {𝑣2, 𝑒2, 𝑒3}. Then 𝐷 β€² is a 𝛾 β€²-set of 𝐺. Since 𝛾(𝐺) = 3, we have𝛾 β€²(𝐺) = 3. Let 𝑛 = 4.Let 𝐷 = {𝑣1, 𝑣4}. Then 𝐷 is a 𝛾-set of 𝐺 so that 𝛾(𝐺)) = 2. Let 𝐷 β€² = {𝑣2, 𝑣3, 𝑒3}. Then 𝐷 β€² is a 𝛾 β€²-set of 𝐺 and so𝛾 β€²(𝐺) ≀ 3. It is easily observed that no two element subsets of 𝐺 is not a 𝛾 β€²-set of 𝐺. Therefore 𝛾 β€²(𝐺)) = 3. Let 𝑛 β‰₯ 5. Let 𝐷 = {𝑣1, 𝑣4}. Then 𝐷 is a dominatingset of 𝐺 and so 𝛾(𝐺) = 2. Let 𝐷′ = {𝑣2, 𝑣5}. Then 𝐷 β€² is a inverse dominating set of 𝐺 and so 𝛾 β€²(𝐺) = 2. ∎ Theorem 2.8. For the graph 𝐺 = 𝐽(𝐹𝑛 ) (𝑛 β‰₯ 5), 𝛾𝑛𝑠 β€² (𝐺) = { 3 𝑖𝑓 𝑛 = 3 2 𝑖𝑓 𝑛 β‰₯ 5 . Proof. The number of vertices of fan graph is 𝑛 + 1. Then it has 2𝑛 βˆ’ 1 edges. Let 𝐺 = 𝐽(𝐹𝑛 ). The number of vertices of jump graph of the fan graph is 2𝑛 βˆ’ 1. Let the vertices of the graph is labeled as {𝑒1, 𝑒2, … , 𝑒𝑛, 𝑣1, 𝑣2, … , π‘£π‘›βˆ’1}, Since 𝐺 contains no universal vertices, 𝛾(𝐺) β‰₯ 2. Let 𝑛 = 3. It is easily verified that no two element subsets of 𝐽(𝐹𝑛 ) is not a 𝛾-set of 𝐺 and so 𝛾(𝐺) β‰₯ 3. Let 𝐷 = {𝑣2, 𝑒2, 𝑒3}. Then 𝐷 is a 𝛾-set of 𝐺 so that 𝛾(𝐺) = 3. Let 𝐷′ = {𝑣1, 𝑣3, 𝑒4}. Then 𝐷 β€² is a non-split inverse dominating set of 𝐺 so that 𝛾𝑛𝑠 β€² (𝐺) = 3. Let 𝑛 = 4. Let 𝐷 = {𝑣1, 𝑣4}. Then 𝐷 is a 𝛾-set of 𝐺 so that 𝛾(𝐺) = 2. Let 𝐷 β€² = {𝑣2, 𝑣3, 𝑒3}. Then 𝐷 β€² is a not a non-split inverse dominating set of 𝐺. Let 𝑛 β‰₯ 5. Let 𝐷 = {𝑣1, 𝑣4}. Then 𝐷 is a dominatingset of 𝐺 and so 𝛾(𝐺) = 2. Let 𝐷′ = {𝑣2, 𝑣5}. Then 𝐷 β€² is a non-split inverse dominating set of 𝐺 and so 𝛾𝑛𝑠 β€² (𝐺) = 2. ∎ Theorem 2.9. For the graph 𝐺 = 𝐽(𝐹𝑛 ) (𝑛 β‰₯ 5), 𝛾𝑐 β€²(𝐺) = { 3 𝑖𝑓 𝑛 = 3 2 𝑖𝑓 𝑛 β‰₯ 5 . Proof. The number of vertices of fan graph is 𝑛 + 1. Then it has 2𝑛 βˆ’ 1 edges. Let 𝐺 = 138 Inverse Domination Parameters of Jump 𝐽(𝐹𝑛 ). The number of vertices of jump graph of the fan graph is 2𝑛 βˆ’ 1. Let the vertices of the graph is labeled as {𝑒1, 𝑒2, … , 𝑒𝑛, 𝑣1, 𝑣2, … , π‘£π‘›βˆ’1}, Since 𝐺 contains no universal vertices, 𝛾(𝐺) β‰₯ 2. Let 𝑛 = 3. It is easily verified that no two element subsets of 𝐽(𝐹𝑛 ) is not a 𝛾-set of 𝐺 and so 𝛾(𝐺) β‰₯ 3. Let 𝐷 = {𝑣2, 𝑒2, 𝑒3}. Then 𝐷 is a 𝛾-set of 𝐺 so that 𝛾(𝐺) = 3. Let 𝐷′ = {𝑣1, 𝑣3, 𝑒4}. Then 𝐷 β€² is a connected inverse dominating set of 𝐺 so that 𝛾𝑐 β€²(𝐺) = 3. Let 𝑛 = 4. Let 𝐷 = {𝑣1, 𝑣4}. Then 𝐷 is a 𝛾-set of 𝐺 so that 𝛾(𝐺) = 2. Let 𝐷 β€² = {𝑣2, 𝑣3, 𝑒3}. Then 𝐷 β€² is a not a connected inverse dominating set of 𝐺. Let 𝑛 β‰₯ 5. Let 𝐷 = {𝑣1, 𝑣4}. Then 𝐷 is a dominatingset of 𝐺 and so 𝛾(𝐺) = 2. Let 𝐷′ = {𝑣2, 𝑣5}. Then 𝐷 β€² is a connected inverse dominating set of 𝐺so that𝛾𝑐 β€²(𝐺) = 2. ∎ Inverse domination number of jump graph of π‘ͺ𝒏 βŠ™ π‘²πŸ Theorem 2.10. For the graph 𝐺 = 𝐽(𝐢𝑛 βŠ™ 𝐾2) (𝑛 β‰₯ 4), 𝛾 β€²(𝐺) = { 3 𝑖𝑓 𝑛 = 3 2 𝑖𝑓 𝑛 β‰₯ 4 . Proof. The number of vertices of 𝐢𝑛 βŠ™ 𝐾2is 3𝑛. Then it has 3𝑛 edges. Let 𝐺 = 𝐽(𝐢𝑛 βŠ™ 𝐾2). The number of vertices of jump graph of the 𝐢𝑛 βŠ™ 𝐾2 is 3𝑛. Let the vertices of the graph is labeled as {𝑒1, 𝑒2, … , 𝑒𝑛, 𝑣1, 𝑣2, … , 𝑣𝑛 , 𝑀1, 𝑀2, … , 𝑀𝑛}. Since 𝐺 contains no universal vertices, 𝛾(𝐺) β‰₯ 2. Let 𝑛 = 3. It is easily verified that no two element subsets of 𝐽(𝐢𝑛 βŠ™ 𝐾2) is not a 𝛾-set of 𝐺 and so 𝛾(𝐺) β‰₯ 3. Let 𝐷 = {𝑣2, 𝑀2, 𝑀3}. Then 𝐷 is a 𝛾-set of 𝐺 so that 𝛾(𝐺) = 3. Let 𝐷′ = {𝑒1, 𝑒2, 𝑒3}. Then 𝐷 β€² is a 𝛾 β€²-set of 𝐺. Since 𝛾(𝐺) = 3, we have 𝛾 β€²(𝐺) = 3. Let 𝑛 β‰₯ 4. Let 𝐷 = {𝑒2, 𝑒4}. Then 𝐷 is a dominatingset of 𝐺 and so 𝛾(𝐺) = 2. Let 𝐷′ = {𝑀2, 𝑀4}. Then 𝐷 β€² is a inverse dominating set of 𝐺 so that𝛾 β€²(𝐺) = 2. ∎ Theorem 2.11. For the graph 𝐺 = 𝐽(𝐢𝑛 βŠ™ 𝐾2) (𝑛 β‰₯ 4), 𝛾𝑛𝑠 β€² (𝐺) = { 3 𝑖𝑓 𝑛 = 3 2 𝑖𝑓 𝑛 β‰₯ 4 . Proof. The number of vertices of 𝐢𝑛 βŠ™ 𝐾2 is 3𝑛. Then it has 3𝑛 edges. Let 𝐺 = 𝐽(𝐢𝑛 βŠ™ 𝐾2). The number of vertices of jump graph of the 𝐢𝑛 βŠ™ 𝐾2 is 3𝑛. Let the vertices of the graph is labeled as {𝑒1, 𝑒2, … , 𝑒𝑛, 𝑣1, 𝑣2, … , 𝑣𝑛 , 𝑀1, 𝑀2, … , 𝑀𝑛}. Since 𝐺 contains no universal vertices, 𝛾(𝐺) β‰₯ 2. Let 𝑛 = 3. It is easily verified that no two element subsets of 𝐽(𝐢𝑛 βŠ™ 𝐾2) is not a 𝛾-set of 𝐺 and so 𝛾(𝐺) β‰₯ 3. Let 𝐷 = {𝑣2, 𝑀2, 𝑀3}. Then 𝐷 is a 𝛾-set of 𝐺 so that 𝛾(𝐺) = 3. Let 𝐷′ = {𝑒1, 𝑒2, 𝑒3}. Then 𝐷 β€² is a non-split inverse dominating set of 𝐺 so that 𝛾𝑛𝑠 β€² (𝐺) = 3. Let 𝑛 β‰₯ 4. Let 𝐷 = {𝑒2, 𝑒4}. Then𝐷 is a dominating set of 𝐺 so that 𝛾(𝐺) = 2. Let𝐷 β€²= {𝑀2, 𝑒4}. Then 𝐷 β€² is a non-split inverse dominating set of 𝐺 so that 𝛾𝑛𝑠 β€² (𝐺) = 2. Theorem 2.12. For the graph 𝐺 = 𝐽(𝐢𝑛 βŠ™ 𝐾2) (𝑛 β‰₯ 4), 𝛾𝑐 β€²(𝐺) = { 3 𝑖𝑓 𝑛 = 3 2 𝑖𝑓 𝑛 β‰₯ 4 . Proof. The number of vertices of 𝐢𝑛 βŠ™ 𝐾2 is 3𝑛. Then it has 3𝑛 edges. Let 𝐺 = 139 S. Santha and G.T. Krishna Veni 𝐽(𝐢𝑛 βŠ™ 𝐾2). The number of vertices of jump graph of the 𝐢𝑛 βŠ™ 𝐾2 is 3𝑛. Let the vertices of the graph is labeled as {𝑒1, 𝑒2, … , 𝑒𝑛, 𝑣1, 𝑣2, … , 𝑣𝑛 , 𝑀1, 𝑀2, … , 𝑀𝑛}. Since 𝐺 contains no universal vertices, 𝛾𝑐 β€²(𝐺) β‰₯ 2. Let 𝑛 = 3. It is easily verified that no two element subsets of 𝐽(𝐢𝑛 βŠ™ 𝐾2) is not a 𝛾-set of 𝐺 and so 𝛾(𝐺) β‰₯ 3. Let 𝐷 = {𝑣2, 𝑀2, 𝑀3}. Then 𝐷 is a 𝛾-set of 𝐺 so that 𝛾(𝐺) = 3. Let 𝐷′ = {𝑒1, 𝑒2, 𝑒3}. Then 𝐷 β€² is a non-split inverse dominating set of 𝐺 so that 𝛾𝑐 β€²(𝐺) = 2. Let 𝑛 β‰₯ 4. Let 𝐷 = {𝑒2, 𝑒4}. Then 𝐷 is a dominating set of 𝐺 so that 𝛾(𝐺) = 2. Let 𝐷′ = {𝑀2, 𝑀4}. Then 𝐷 β€² is a connected inverse dominating set of 𝐺 so that 𝛾𝑐 β€²(𝐺) = 2. ∎ Inverse domination number of jump graph of 𝑷𝒏 βŠ™ π‘²πŸ Theorem 2.13. For the graph 𝐺 = 𝐽(𝑃𝑛 βŠ™ 𝐾2) (𝑛 β‰₯ 3), 𝛾 β€²(𝐺) = 2. Proof. For𝑛 β‰₯ 3,the number of vertices of 𝑃𝑛 βŠ™ 𝐾2is 3𝑛. Then it has 3𝑛 βˆ’ 1 edges. Let 𝐺 = 𝐽(𝑃𝑛 βŠ™ 𝐾2). The number of vertices of jump graph of the 𝑃𝑛 βŠ™ 𝐾2 is 3𝑛 βˆ’ 1. Let the vertices of the graph is labeled as {𝑒1, 𝑒2, … , 𝑒𝑛 , 𝑣1, 𝑣2, … π‘£π‘›βˆ’1, 𝑀1, 𝑀2, …, 𝑀𝑛}. Since 𝐺 contains no universal vertices, 𝛾(𝐺) β‰₯ 2. Let 𝐷 = {𝑒1, 𝑒3}. Then 𝐷 is a 𝛾- set of 𝐺 so that 𝛾(𝐺) = 2. Let 𝐷′ = {𝑀1, 𝑀3}. Then 𝐷 β€² is a inverse dominating set of 𝐺 so that𝛾 β€²(𝐺) = 2. ∎ Theorem 2.14. For the graph 𝐺 = 𝐽(𝑃𝑛 βŠ™ 𝐾2) (𝑛 β‰₯ 3), 𝛾𝑛𝑠 β€² (𝐺) = 2. Proof. For𝑛 β‰₯ 3,the number of vertices of 𝑃𝑛 βŠ™ 𝐾2is 3𝑛. Then it has 3𝑛 βˆ’ 1 edges. Let 𝐺 = 𝐽(𝑃𝑛 βŠ™ 𝐾2). The number of vertices of jump graph of the 𝑃𝑛 βŠ™ 𝐾2 is 3𝑛 βˆ’ 1. Let the vertices of the graph is labeled as {𝑒1, 𝑒2, … , 𝑒𝑛 , 𝑣1, 𝑣2, … , π‘£π‘›βˆ’1, 𝑀1, 𝑀2, … , 𝑀𝑛}. Since 𝐺 contains no universal vertices, 𝛾(𝐺) β‰₯ 2. Let 𝐷 = {𝑒1, 𝑒3}. Then 𝐷 is a dominating set of 𝐺and so 𝛾(𝐺) = 2. Let 𝐷′ = {𝑀1, 𝑀3}. Then 𝐷 β€² is a non-split inverse dominating set of 𝐺 so that 𝛾𝑛𝑠 β€² (𝐺) = 2. ∎ Theorem 2.15. For the graph 𝐺 = 𝐽(𝑃𝑛 βŠ™ 𝐾2) (𝑛 β‰₯ 3), 𝛾𝑐 β€²(𝐺) = 2. Proof. For𝑛 β‰₯ 3,the number of vertices of 𝑃𝑛 βŠ™ 𝐾2is 3𝑛. Then it has 3𝑛 βˆ’ 1 edges. Let 𝐺 = 𝐽(𝑃𝑛 βŠ™ 𝐾2). The number of vertices of jump graph of the 𝑃𝑛 βŠ™ 𝐾2 is 3𝑛 βˆ’ 1. Let the vertices of the graph is labeled as{𝑒1, 𝑒2, … , 𝑒𝑛 , 𝑣1, 𝑣2, … , π‘£π‘›βˆ’1, 𝑀1, 𝑀2, …, , 𝑀𝑛}. Since 𝐺 contains no universal vertices, 𝛾(𝐺) β‰₯ 2. Let 𝐷 = {𝑒1, 𝑒3}. Then 𝐷 is a dominating set of 𝐺 and so 𝛾(𝐺) = 2. Let 𝐷′ = {𝑀1, 𝑀3}. Then 𝐷 β€² is a connected inverse dominating set of 𝐺 so that 𝛾𝑐 β€²(𝐺) = 2. ∎ 3. Conclusions In this article, we determined some inverse domination parameter for jump graph of some special graphs. We will determine some more inverse domination parameters for jump graph of some special graph in future work. 140 Inverse Domination Parameters of Jump References [1] Allan R.B. and R. C. Laskar, On domination inverse domination numbers of a graph, Discrete Math.,23, ,73-76, 1978. [2] J.A. Bondy, U.S.R. Murty, Graph Theory with Applications, North Holland, New York. [3] G. Chatrand, H. Galvas, K.C, Vandell and F. Harary, The Forcing Domination Number of a graph, J. Combin. Math, Combin. Comput., (1997), 25: 161-174. [4] E.J. Cockayne and Hedetniemi., Towards a theory of domination in graphs, Networks, 7, 247-261, 1997. [5] T.W. Haynes and S.T. Hedetniemi, P.J. Slater, Fundamentals of Domination in Graph, Marcel Dekkar, New York,1998. [6] J. John, The forcing monophonic and the forcing geodetic numbers of a graph, Indonesian Journal of Combinatorics, 4(2), (2020),114-125. [7] J. John and Malchijah Raj, The forcing non-split domination number of a graph, Korean Journal of Mathematics, 29(1) (2021),1-12. [8] J. John and V. Sujin Flower, On the forcing domination and forcing total domination number of a graph, Graphs and Combintorics, 38(5), (2022). https://doi.org/10.1007/s00373-022-02521-y [9] M. Karthikeyan and A. Elumalai, Inverse Domination Number of Jump Graph. Intn. joum. pure and applied maths vol 103 No.3, 477-483 100 – 102, 2015. [10] V.R. Kulli and S.C. Sigarkanthi, Inverse Domination in Graph, Nat. Acad. Sci. Letters, 14, 473-475, 1991. [11] Y.B. Maralabhavi, S.B. Anupama and Venkanagouda M. Goudar, Domination Number of Jump Graph, Intn. Mathematical Forum, vol8, 753-758, 2013. [12] Dr. S. Meena and K. Vaithilingan, Prime Labeling For Some Fan Related Graphs, International Journal of Engineering Research and Technology (IJERT), 9, 141