# The steiner tree problem and homogeneous sets

## Abstract

The Steiner tree problem has been extensively studied in the literature because of its various applications (e.g., network design, circuit layout and database query answering). The main result in the paper states that an instance of this problem on a graph *G* is polynomially reducible to an instance of the same problem on an induced subgraph of *G* whenever *G* contains a *homogeneous set* of nodes (i.e., a set of two or more nodes such that every reamining node in *G* is adjacent either to all or to none of its nodes). This result allows, in particular, to solve in polynomial time the Steiner tree problem on a new class of graphs, called *homogeneous graphs*, defined in terms of homogeneous sets, that properly contains some classes of graphs on which the problem is polynomially solvable. A polynomial algorithm to recognize homogeneous graphs is also given.

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