Malaya Journal of Matematik, Vol. S, No. 1, 465-468, 2020 https://doi.org/10.26637/MJM0S20/0086 A note on the double domination in ladder graph Suhas P. Gade 1 * Abstract For any graph G =( V, E ) the ladder graph L n is defined by L n = P n × K 2 where P n is a path with n vertices and × denotes the Cartesian product and K 2 is a complete graph with two vertices. A set D d of V [L n ] is double dominating set of L n if for every vertex v ∈ V [L n ], | N[v] ∩ D d |= 2, that is v is in D d and has at least one neighbor in D d or v is in V [L n ] - D d and has at least two neighbors in D d . The double domination number γ dd (L n ) in ladder graph L n is a minimum cardinality of double dominating set. In this paper many sharp bounds on γ dd (L n ) are obtained and its exact value for ladder graph were found in terms of parameter of G. Also its relationship with other domination parameters is investigated. Keywords Domination number, (1,2)-Domination number, Upper Pendant domination number, Double domination number. AMS Subject Classification 05C69. 1 Department of Mathematics, Sangameshwar College, Solapur-413001, Maharashtra, India. *Corresponding author: 1 suhaspanduranggade@gmail.com Article History: Received 10 January 2020; Accepted 01 May 2020 ©2020 MJM. Contents 1 Introduction ....................................... 465 2 Main Results ...................................... 466 3 Relation Between Domination Number, (1,2)- Domi- nation Number, Upper Pendent Domination Number and Double Domination Number of Ladder Graph 467 4 Algorithm for to Find a Minimal Double Dominating Set ................................................. 467 5 Conclusion ........................................ 467 References ........................................ 467 1. Introduction Let G =( V, E ) be a simple graph. The open neighbour- hood N(v) of the vertex v consists of the set of vertices ad- jacent to v, that is, N(v)= {w ∈ V : vw ∈ E }, and the closed neighbourhood of v is N[v]= N(v) ∪{v}. A set S ⊆ V of vertices in a graph G =( V, E ) is called a dominating set if every vertex v ∈ V is either an element of S or is adjacent to an element of S. The domination number of G, denoted by γ (G), is the minimum cardinality of a dominating set of G. A (1,2)-dominating set in a graph G =( V, E ) is a set S such that for any vertex v ∈ V \ D there is atleast one vertex in S at distance 1 from v and a second vertex in S at distance at most 2 from v. The order of the smallest (1,2)-dominating set of Gis called the (1,2)-domination number of G and we denoted it by γ (1,2) (G). A dominating set D in G is called a pendant dominating set if < D > contains at least one pendant vertex. The minimum cardinality of a pendant dominating set is called the pendent domination number denoted by γ pe (G). The minimal pendant dominating set with maximum cardinality is called the upper pendant dominating set and is denoted by Γ pe (G). A set D d of V [G] is double dominating set of G if for each vertex v ∈ V [G], |N[v] ∩ D d |≥ 2, that is v is in D d and has at least one neighbor in D d or v is in V [G] - D d and has at least two neighbors in D d . The size of a smallest double dominating set is called the double domination number of G and it is denoted by γ dd (G). A Cartesian product of two graph G and H is the graph K = G × H has V (K)= V (G) × V (H) and vertices (u 1 , v 1 ) and (u 2 , v 2 ) in V (K) are adjacent if and only if either u 1 = u 2 and v 1 v 2 ∈ E (H) or v 1 = v 2 and u 1 u 2 ∈ E (G). The ladder graph L n is defined by L n = P n × K 2 where P n is a path with n vertices and × denotes the Cartesian product and K 2 is a complete graph with two vertices. For notation and terminology we follow [1, 4]. For the survey on the area of domination in ladder graph have been considered and investigated in [5, 6, 9–12]. The concepts of double domination in graphs with its