On balance and consistency preserving 2-path signed graphs

Abstract

Let Σ = (G, σ) be a balanced and canonically consistent signed graph. The 2-path signed graph Σ#Σ = (G 2 , σ ′ ) of Σ has the underlying graph as G 2 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, G 2 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.