A NOTE ON THE DOUBLE DOMINATION NUMBER IN MAXIMAL OUTERPLANAR AND PLANAR GRAPHS

. In a graph, a vertex dominates itself and its neighbors. A subset 𝑆 of vertices of a graph 𝐺 is a double dominating set of 𝐺 if 𝑆 dominates every vertex of 𝐺 at least twice. The double domination number 𝛾 × 2 ( 𝐺 ) of 𝐺 is the minimum cardinality of a double dominating set of 𝐺 . In this paper, we prove that the double domination number of a maximal outerplanar graph 𝐺 of order 𝑛 is bounded above by 𝑛 + 𝑘 2 , where 𝑘 is the number of pairs of consecutive vertices of degree two and with distance at least 3 on the outer cycle. We also prove that 𝛾 × 2 ( 𝐺 ) ≤ 5 𝑛 8 for a Hamiltonian maximal planar graph 𝐺 of order 𝑛 ≥ 7.


Introduction
For graph theory notation and terminology not given here we refer to [6].We consider finite, undirected and simple graphs  with vertex set  =  () and edge set  = ().The number of vertices of  is called the order of  and is denoted by  = ().The open neighborhood of a vertex  ∈  is  () =   () = { ∈  |  ∈ } and the closed neighborhood of  is  [] =   [] =  () ∪ {}.The degree of a vertex , denoted by deg() (or deg  () to refer to ), is the cardinality of its open neighborhood.We denote by () and ∆(), the minimum and maximum degrees among all vertices of , respectively.A plane graph  is said to be a triangulated disc if all of its faces except the infinite face are triangles.A graph  is outerplanar if it has an embedding in the plane such that all vertices belong to the boundary of its outer face.A planar (resp.outerplanar) graph  is maximal if  +  is not planar (resp.outerplanar) for any two nonadjacent vertices  and  of .An inner face of a maximal outerplanar graph  is said to be an internal triangle if it is not adjacent to the outer face.A maximal outerplanar graph  is called striped if it has no internal triangles.A subset  ⊆  is a dominating set of  if every vertex in  − has a neighbor in .The domination number () is the minimum cardinality of a dominating set of .For a comprehensive survey on the subject of domination parameters in graphs the reader can refer to the two books [6,7].
Harary and Haynes [5] defined a generalization of domination, namely -tuple domination.For a positive integer , a subset  of vertices of a graph  is a -tuple dominating set of  if for every vertex  ∈  (), | [] ∩ | ≥ .The -tuple domination number  × () is the minimum cardinality of a -tuple dominating set of , if such a set exists.A -tuple dominating set where  = 2 is called a double dominating set.A double dominating set of cardinality  ×2 () is referred to as a  ×2 ()-set.The concept of double domination in graph was further studied in, for example, [1,2,4,8].Blidia et al. [1] showed that  ×2 () ≤ 11 13 if  is a graph of order  with () ≥ 2. Henning [8] proved that  ×2 () ≤ 3 4 provided that  is not a 5-cycle.Domination in maximal planar graphs and outer-planar graphs has recieved great attention and several domination parameters for these classes of graph have been studied.See, for example, Dorfling et al. [3], Henning and Kaemawichanurat [9], Lemanska et al. [11], Li et al. [12], King and Pelsmajer [10], Matheson and Tarjan [14], Tokunaga [16] and Liu [13].Recently, Zhuang [18] studied double domination in maximal outerplanar graphs, and proved the following.
Theorem 1.2 (Zhuang [18]).Let  be a maximal outerplanar graph of order  ≥ 3 and  be the number of vertices of degree 2 in .
In this paper, we first improve Theorem 1.2 by showing that  ×2 () ≤ + 2 , where  is the number of pairs of consecutive vertices of degree two with distance at least 3 on the outer cycle.We also prove that  ×2 () ≤ 5 8 for a Hamiltonian maximal planar graph  of order  ≥ 7, which improves Theorem 1.1 and all previous bounds.
We follow the notations and method given in [12].For a Hamiltonian maximal planar graph  with a Hamilton cycle , let   in be the maximal outerplanar graph consists of  and all edges inside of  and   out be the maximal outerplanar graph consists of  and all edges outside of .Let  1 , . . .,   be all the vertices of degree 2 which appear in the clockwise direction on .A vertex   is called a bad vertex if the distance between   and  +1 on  is at least 3, for  = 1, 2, . . ., , where the subscript is taken modulo .We make use of the following.
Theorem 1.3 (Li et al. [12]).For a Hamiltonian maximal planar graph  of order , there exists a Hamilton cycle  of  such that   in or   out has at most  4 bad vertices.

Main results
Let  be a maximal outerplanar graph.There is an embedding of  in the plane such that all of its vertices are on the outer cycle  which is the boundary of the outer face and each inner face is a triangle.Let  1 , . . .,   be all the vertices of degree 2 which appear in the clockwise direction on .We will prove the following.

Proof of Theorem 2.2
Let  be a Hamiltonian maximal planar graph of order  ≥ 7. Let  be a Hamilton cycle of , and without loss of generality, assume that   in has at most  4 bad vertices according to Theorem 1.3.Then  ≤  4 and by Theorem 2.1,  ×2 () ≤ + 2 ≤ 5 8 .