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

Abstract

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.