Weighted Maximum-Clique Transversal Sets of Graphs

Algorithm 1

Finding a maximum-clique transversal set of a split graph of minimum weight.

Input: A split graph 𝐺=(𝐼∪𝑄,𝐸,𝑤).

Output: A maximum-clique transversal set 𝐷 of 𝐺 of minimum weight.

  (1) 𝑆=𝐼;

  (2) for each vertex 𝑣∈𝑄  do

  (3) 𝑤_𝑁(𝑣)=𝑤(𝑁_𝐺(𝑣));

  (4) end for

  (5) for each vertex 𝑠∈𝐼  do

  (6)  if   deg_𝐺(𝑠)<𝜔(𝐺)−1  then

  (7)   𝑆←𝑆⧵{𝑠};

  (8)   for each 𝑣∈𝑁_𝐺(𝑠)  do

  (9)    𝑤_𝑁(𝑣)=𝑤_𝑁(𝑣)−𝑤(𝑠);

(10)   end for

(11)  end if

(12) end for

(13) Find a vertex 𝑣₁∈𝑄 such that 𝑤(𝑣₁)=min{𝑤(𝑣)∣𝑣∈𝑄};

(14) Find a vertex 𝑣₂∈𝑄⧵{𝑣₁} such that 𝑤(𝑣₂)=min{𝑤(𝑣)∣𝑣∈𝑄⧵{𝑣₁}};

(15) Find a vertex 𝑣₃∈𝑄 such that 𝑤(𝑆∪𝑄)−𝑤_𝑁(𝑣₃)=min{𝑤(𝑆∪𝑄)−𝑤_𝑁(𝑣)∣𝑣∈𝑄};

(16) if min{𝑤(𝑣₁)+𝑤(𝑣₂),𝑤(𝑆∪𝑄)−𝑤_𝑁(𝑣₃)}=𝑤(𝑣₁)+𝑤(𝑣₂)  then

(17)  𝐷={𝑣₁,𝑣₂};

(18) else

(19)  𝐷={𝑣₃}∪(𝑆⧵𝑁_𝐺(𝑣₃));

(20) end if

(21) Output the set 𝐷;