Detecting an induced net subdivision

TL;DR

This paper presents a polynomial-time algorithm to detect whether a graph contains an induced subgraph that is a subdivision of a net, a specific graph structure involving a triangle and three pendant vertices.

Contribution

It introduces a novel polynomial-time algorithm for identifying induced net subdivisions, a problem not solvable by existing three-in-a-tree methods.

Findings

Algorithm successfully detects induced net subdivisions in polynomial time.

The approach differs from previous methods by not relying on three-in-a-tree routines.

Provides a new tool for graph structure detection in polynomial time.

Abstract

A {\em net} is a graph consisting of a triangle $C$ and three more vertices, each of degree one and with its neighbour in $C$, and all adjacent to different vertices of $C$. We give a polynomial-time algorithm to test whether an input graph has an induced subgraph which is a subdivision of a net. Unlike many similar questions, this does not seem to be solvable by an application of the "three-in-a-tree" subroutine.