Steiner Tree in $k$-star Caterpillar Convex Bipartite Graphs -- A Dichotomy

TL;DR

This paper investigates the computational complexity of the Steiner Tree problem in a new class of bipartite graphs called $k$-star caterpillar convex bipartite graphs, establishing NP-completeness for certain subclasses and polynomial-time solvability for others.

Contribution

It introduces $k$-star caterpillar convex bipartite graphs and determines the complexity of STREE within these classes, providing a clear dichotomy.

Findings

STREE is NP-complete for 1-star caterpillar convex bipartite graphs.

STREE is polynomial-time solvable for 0-star (convex bipartite) graphs.

Strengthens previous NP-completeness results for chordal bipartite graphs.

Abstract

The class of $k$-star caterpillar convex bipartite graphs generalizes the class of convex bipartite graphs. 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 $k$-star caterpillar on $X$ is imaginary and if the imaginary structure is a path ($0$-star caterpillar), then it is the class of convex bipartite graphs. 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. STREE is known to be NP-complete on general graphs as well as for special graph classes such as chordal graphs, bipartite graphs, and chordal bipartite graphs. The complexity of STREE in convex bipartite graphs, which is a popular subclass of chordal bipartite graphs, is open. In this paper, we introduce $k$-star caterpillar convex bipartite graphs, and 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). In \cite{muller1987np}, it is shown that STREE in chordal bipartite graphs is NP-complete. A close look at the reduction instances reveal that the instances are $3$-star caterpillar convex bipartite graphs, and in this paper, we strengthen the result of \cite{muller1987np}.