# Details of talk

Title The switching-stable graphs Jeanette McLeod (University of Canterbury) Jeanette McLeod, Brendan McKay Algebra and Discrete Mathematics 14:30:00 2017-09-25  The classical reconstruction problem asks when a graph $G$ can be reconstructed from its \emph{deck}, where the deck consists of \emph{cards} showing each of the vertex-deleted subgraphs of~$G$. Stanley, Bondy and others introduced variants of this problem, where the cards instead show the graphs obtained by \emph{switching}~$G$. While investigating switching reconstruction problems, Bondy asked which graphs have the property that every possible switching produces a graph that is equivalent to the original graph. Such graphs are said to be \emph{switching-stable}. We answer Bondy's question for two types of switching. First, we let the edges of $G$ be coloured with two colours and define the switching operation to be the interchange of the colours of the edges incident to a specified vertex and classify all switching-stable 2-edge-coloured graphs. Then, we let $G$ be a digraph and define the switching operation to be the reversal of the directions of the edges incident to a specified vertex and classify all switching-stable digraphs. For both variants of switching, we consider the problem for two different types of equivalence.