Dominating sets in finite generalized quadrangles - Héger, Tamás (Ars mathematica contemporanea, 2020)
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 ?$2st+1$?, 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.
Keywords: dominating set; dominirajoča množica; finite generalized quadrangle; končni posplošeni pravokotnik