A Note on $Y–\mathrm{\Delta }$ and $\mathrm{\Delta }–Y$ Graphs

Abstract

A $2$-connected graph is called $Y–\mathrm{\Delta }$ (respectively $\mathrm{\Delta }–Y$) $reducible$ or simply a $Y–\mathrm{\Delta }$ (respectively $\mathrm{\Delta }–Y$) graph if it can be reduced to a single edge using a sequence of $Y–\mathrm{\Delta }$ (respectively $\mathrm{\Delta }–Y$, series and parallel reductions. This paper addresses the problem of decomposing $Y–\mathrm{\Delta }$ and $\mathrm{\Delta }–Y$ graphs in connection with a new method for decomposing $3$-connected graphs proposed recently by Coullard, Gardner, and Wagner.