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.