All induced connected subgraphs of a graphG contain a dominating set of pair-wise adjacent vertices if and only if there is no induced subgraph isomorphic to a path
or a cycle of five vertices inG. Moreover, the problem of finding any given type of connected dominating sets in all members of a classG of graphs can be reduced to the graphsG∈G that have a cut-vertex or do not contain any cutsetS dominated by somes∈S.
In this paper we consider three problems concerning systems of vector exponentials. In the first part we prove a conjecture of V. Komornik raised in  on the independence of the movement of a rectangular membrane in different points. It was independently proved by M. Horváth  and S. A. Avdonin (personal communication). The analogous problem for the circular membrane was partly solved in  — the complete solution is given in . In the second part we fill in a gap in the theory of Blaschke-Potapov products developed in the paper  of Potapov. Namely we prove that the Blaschke-Potapov product is determined by its kernel sets up to a multiplicative constant matrix. In the third part of the present paper we give a multidimensional generalization of the notion of sine type function developed by Levin ,  and by our generalization we prove the multidimensional variant of the Levin-Golovin basis theorem , .
Josting, A., Rueffer, U., Franklin, J. és mtsai:
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