Characterization of 2-Path Product Signed Graphs with Its Properties

Algorithm 1

To check if the given signed graph is a 2-path of some other signed graph.

Input. The adjacency matrix

Output. If is a 2-path for some signed graph then returns .

Process

(1) Collect all the cliques for each vertex , using Bron-Kerbosch algorithm [33].

(2) Mark every vertex by + and then − in each clique.

(3) Calculate , which represent the all possible combinations generated by each marked vertex from the

   clique.

(4) for to do

(5)   Select

(6)   for to do

(7)     Select

(8)     if   then

(9)       

(10)   for to do

(11)     for to do

(12)       if then

(13)         

(14)       else

(15)         if then

(16)           

(17)         else

(18)           

(19)           

(20)   for to do

(21)     for to do

(22)       if then

(23)         go to (25)

(24)       else

(25)         go to (23)

(26) For all the combinations of elementary swamping operations on either rows or columns in , repeat (4).

(27) If all the combinations are checked and no such matrix is obtained then no such graph exist.

(28) If such a signed graph exist then is the adjacency matrix of required signed graph whose 2-path signed

   graph is .