Exact Solutions for the Steiner Path Cover Problem on Special Graph Classes

The Steiner path problem is a restriction of the well known Steiner tree problem such that the required terminal vertices lie on a path of minimum cost. While a Steiner tree always exists within connected graphs, it is not always possible to find a Steiner path. Despite this, one can ask for the Steiner path cover, i.e. a set of vertex disjoint simple paths which contains all terminal vertices and possibly some of the non-terminal vertices. We show how a Steiner path cover of minimum cardinality for the disjoint union and join composition of two graphs can be computed in linear time from the corresponding values of the involved graphs. The cost of an optimal Steiner path cover is the minimum number of Steiner vertices in a Steiner path cover of minimum cardinality. We compute recursively in linear time the cost within a Steiner path cover for the disjoint union and join composition of two graphs by the costs of the involved graphs. This leads us to a linear time computation of an optimal Steiner path, if it exists, for special co-graphs.