# An Efficient Algorithm for Enumerating Chordal Bipartite Induced Subgraphs in Sparse Graphs

## TL;DR

This paper introduces a new characterization of chordal bipartite graphs, along with an efficient enumeration algorithm for their induced subgraphs, enabling faster analysis of sparse graphs in various applications.

## Contribution

It presents a novel vertex elimination ordering (CBEO) characterization and an algorithm ECB for enumerating chordal bipartite induced subgraphs efficiently.

## Key findings

- Enumeration runs in O(ktΔ^2) time per solution on average.
- ECB achieves constant amortized time for bounded degree graphs.
- The characterization links chordal bipartite graphs to a special vertex ordering.

## Abstract

In this paper, we propose a characterization of chordal bipartite graphs and an efficient enumeration algorithm for chordal bipartite induced subgraphs. A chordal bipartite graph is a bipartite graph without induced cycles with length six or more. It is known that the incident graph of a hypergraph is chordal bipartite graph if and only if the hypergraph is $\beta$-acyclic. As the main result of our paper, we show that a graph $G$ is chordal bipartite if and only if there is a special vertex elimination ordering for $G$, called CBEO. Moreover, we propose an algorithm ECB which enumerates all chordal bipartite induced subgraphs in $O(kt\Delta^2)$ time per solution on average, where $k$ is the degeneracy, $t$ is the maximum size of $K_{t,t}$ as an induced subgraph, and $\Delta$ is the degree. ECB achieves constant amortized time enumeration for bounded degree graphs.