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Abstract: A graph is dot-critical if contracting any edge decreases the domination number. It is totally dot-critical if identifying any two vertices decreases the domination number. Burton and Sumner [Discrete Math. 306 (2006) 11–18] posed the problem: Is it true that for , there exists a totally -dot-critical graph with no critical vertices? In this paper, we show that this problem has a positive answer. [Copyright &y& Elsevier]

Abstract: In [On Mills''s conjecture on matroids with many common bases, Discrete Math. 240 (2001) 271–276], Lemos proved a conjecture of Mills [On matroids with many common bases, Discrete Math. 203 (1999) 195–205]: for two -connected matroids whose symmetric difference between their collections of bases has size at most , there is a matroid that is obtained from one of these matroids by relaxing circuit-hyperplanes and from the other by relaxing circuit-hyperplanes, where and are non-negative integers such that . In [Matroids with many common bases, Discrete Math. 270 (2003) 193–205], Lemos proved a similar result, where the hypothesis of the matroids being -connected is replaced by the weaker hypothesis of being vertically -connected. In this paper, we extend these results. [Copyright &y& Elsevier]