Dominating sets in finite generalized quadrangles

Abstract

A dominating set in a graph is a set of vertices such that each vertex not in the set has a neighbor in the set. The domination number is the smallest size of a dominating set. We consider this problem in the incidence graph of a generalized quadrangle. We show that the domination number of a generalized quadrangle with parameters s and t is at most 2 s t  + 1 , and we prove that this bound is sharp if s  =  t or if s  =  q  − 1 and t  =  q  + 1 . Moreover, we give a complete classification of smallest dominating sets in generalized quadrangles where s  =  t , and give some general results for small dominating sets in the general case.