Some Steiner concepts on lexicographic products of graphs

The smallest tree that contains all vertices of a subset W of V (G) is called a Steiner tree. The number of edges of such a tree is the Steiner distance of W and union of all Steiner trees of W form a Steiner interval. Both of them are described for the lexicographic product in the present work. We also give a complete answer for the following invariants with respect to the Steiner convexity: the Steiner number, the rank, the hull number, and the Carathéodory number, and a partial answer for the Radon number. At the end we locate and repair a small mistake from [7]. ∗Work supported by the Ministry of Science of Slovenia and by the Ministry of Science and Technology of India under the bilateral India-Slovenia grants BI-IN/10-12-001 and INT/SLOVENIA-P17/2009, respectively. †Supported by the Ministry of Science of Slovenia under the grant P1-0297. The author is also with IMFM, Jadranska 19, 1000 Ljubljana, Slovenia. ‡Supported by the University Grants Commission, Govt. of India under the grant MRP(S)864/10-11/KLCA042/UGC-SWRO. 1 Pr ep ri n t se ri es , I M FM , I S S N 2 23 220 94 , n o. 1 17 9, J u ly 3 8, 2 01 2