The definition of the Workday Number of a finite graph is given on page 14 in http://www.arml.com/2012_contest/2012_Contest_Final_Version.pdf and the rest of the problem statement is given at the top of page 12 (omitted here for brevity). Is the general problem of computing the Workday Number of an arbitrary graph based on a well-known area of graph theory, perhaps under another name? Are there references that would show how results about the Workday Number of an arbitrary finite graph fit into the current research literature in graph theory? i.e. if such results for arbitrary graphs are new or already known?

I coach an exceptionally talented ARML student who has some nice general results about Workday Numbers that go well beyond the contest questions and would like to advise him on how to best write up his results.