On the radius and the attachment number of tetravalent half-arc-transitive graphs

In this paper, we study the relationship between the radius r and the attachment number a of a tetravalent graph admitting a half-arc-transitive group of automorphisms. These two parameters were first introduced in Marušič (1998), where among other things it was proved that a always divides 2r. Intrigued by the empirical data from the census (Potočnik et al., 2015) of all such graphs of order up to 1000 we pose the question of whether all examples for which a does not divide r are arc-transitive. We prove that the answer to this question is positive in the case when a is twice an odd number. In addition, we completely characterise the tetravalent graphs admitting a half-arc-transitive group with r=3 and a=2, and prove that they arise as non-sectional split 2-fold covers of line graphs of 2-arc-transitive cubic graphs.