On balance and consistency preserving 2-path signed graphs

Kshittiz Chettri (Department of Mathematics, Nar Bahadur Bhandari Govt. College, Tadong, Sikkim, India)
Biswajit Deb (Department of Mathematics, Sikkim Manipal Institute of Technology, Sikkim Manipal University, Majitar, Sikkim, India.)
Anjan Gautam (Department of Mathematics, Sikkim Manipal Institute of Technology, Sikkim Manipal University, Majitar, Sikkim, India, 737132)

Electronic Journal of Graph Theory and Applications (EJGTA)
Vol 11, No 2 (2023): Electronic Journal of Graph Theory and Applications

Let Σ = (G, σ) be a balanced and canonically consistent signed graph. The 2-path signed graph Σ#Σ = (G2, σ′) of Σ has the underlying graph as G2 and the sign σ′(uv) of an edge uv in it is −1 whenever in each uv-path of length 2 in Σ all edges are negative; otherwise σ′(uv) is 1. Here, G2 is the graph obtained from G by adding an edge between u and v if there is a path of length 2 between them. In this article, we have investigated balancedness and canonically consistency of 2-path signed graphs Σ#Σ of a balanced and canonically consistent signed graph Σ. The problem has been resolved completely for cycles, star graphs and trees.