Minimum 2-Tuple Dominating Set of an Interval Graph

Algorithm 1

Algorithm  2DIG.

Input: An interval graph 𝐺=(𝑉,𝐸) with IG ordering vertex set 𝑉={1,2,3,…,𝑛}.

Output: 2-tuple dominating set 𝐷 and 2-tuple domination number 𝛾_×2(𝐺)(=|𝐷|).

Step  1: Set 𝑓(𝑗)=0,∀𝑗=1,2,…,𝑛; //Assume that no vertices are the members

   of 𝐷.//

Step  2: Set 𝑖=1, 𝐷=∅ and 𝐿=∅;

 Step  2.1: Compute 𝑤_𝑖(𝑓)=∑_{𝑣∈𝑁[𝑖]}𝑓(𝑣);

 Step  2.2: If 𝑤_𝑖(𝑓)=0 then //At least the vertex 𝑖 is not adjacent to any of the

          vertices of 𝐷.//

      Set 𝑓(𝑀₀(𝑖))=1 and 𝑓(𝑀₁(𝑖))=1;

       𝐷=𝐷∪{𝑀₀(𝑖)}∪{𝑀₁(𝑖)} and 𝐿=𝐿∪{𝑖};

     else if 𝑤_𝑖(𝑓)=1 then //At least the vertex 𝑖 is connected to one of the

          vertex of 𝐷.//

      If 𝑓(𝑀₀(𝑖))=1 then

       Set 𝑓(𝑀₁(𝑖))=1;

        𝐷=𝐷∪{𝑀₁(𝑖)};

      else

       Set 𝑓(𝑀₀(𝑖))=1;

       𝐷=𝐷∪{𝑀₀(𝑖)};

      end if;

      𝐿=𝐿∪{𝑖};

     else

      Goto Step  2.3;

     end if;

 Step  2.3: Calculate 𝑖=𝑖+1 and go to Step  2.1 and continue until 𝑖>𝑛;

end  2DIG.