Steiner tree in k-star caterpillar convex bipartite graphs: a dichotomy

A bipartite graph G(X, Y) whose vertex set is partitioned into X and Y a convex graph, if there an ordering of $$X=(x_1,\ldots ,x_m)$$ such that for all $$y \in Y$$ , $$N_G(y)$$ consecutive with respect to the X, G said have convexity X. k-star caterpillar tree collection stars each star having k vertices degree one roots are joined by path. For partitions Y, we associate on in its neighborhood induces tree. The minimum Steiner problem (STREE) defined as follows: given connected $$G=(V,E)$$ subset $$R \subseteq V(G)$$ objective find cardinality $$S \cup S$$ subgraph. In this paper, present following dichotomy result: show STREE NP-complete 1-star graphs polynomial-time solvable 0-star (also known graphs). We also strengthen well-known result Müller Brandstädt (Theoret Comput Sci 53(2-3):257-265, 1987), which says chordal (reduction instances 3-star As application, use our results solve: (i) classical dominating graphs, (ii) interval linear time.