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

一个二分图G ( X ,  Y ) 其顶点集被划分为X和Y是一个凸二分图，如果有一个\(X=(x_1,\ldots ,x_m)\)的排序使得对于所有\( y \in Y\)，\(N_G(y)\)关于X的顺序是连续的，并且G相对于X具有凸性。k星毛毛虫是一棵由星星组成的树，每颗星星都有k个一阶顶点，它们的根由一条路径连接。对于具有分区X和Y ，我们在X上关联一个k星毛虫，这样对于Y中的每个顶点，它的邻域都会产生一棵树。最小斯坦纳树问题 (STREE) 定义如下：给定一个连通图\(G=(V,E)\)和一个顶点子集\(R \subseteq V(G)\)，目标是找到最小基数集\(S \subseteq V(G)\)使得集合\(R \cup S\)引出一个连通子图。在本文中，我们提出了以下二分法结果：我们证明 STREE 对于 1 星毛虫凸二部图是 NP 完全的，对于 0 星毛虫凸二部图（也称为凸二部图）是多项式时间可解的。我们还强化了 Müller 和 Brandstädt 的著名结果（Theoret Comput Sci 53(2-3):257-265, 1987），即弦二部图中的 STREE 是 NP 完全的（约简实例是 3 星毛毛虫凸二分图）。作为一个应用程序，我们使用我们的 STREE 结果来解决：（i）凸二部图中的经典支配集问题，（ii）区间图上的 STREE，线性时间。