Publish work extending on PhD with supervisor I recently completed my PhD and have started a position at a different institution. I am writing a paper which extends a model I introduced in my PhD. It wasn't a significant part of my PhD, more of an aside (about 2 pages). The idea of how to extend it came out of a discussion with the external examiner during my PhD defense (my supervisor was not present). I have a good relationship with my PhD supervisor and I know they would be interested in the project. 

I would like to publish this as the sole author. Should my PhD supervisor be a co-author? They have not been involved in (or aware of) the work to extend the model. How do I let them know without potentially affecting our relationship?

Note, I have no reason to believe they would be upset about such things and may be overthinking this...
I would suggest to mask your question by asking for advice, e.g.


  I recently wrote this paper (attach paper) and as I am still relatively new to the academic world and the basic idea for this paper originated during my PhD (see pages X,Y of my thesis), I would like to ask if I should put you as a second author.


If you already send a finished paper, they will most likely understand that you are politely asking for the ok to publish it as sole author, but you are not burning any bridges this way.