A Characterization for the Existence of Connected $f$-Factors of $\textit{ Large}$ Minimum Degree

TL;DR

This paper characterizes when large minimum degree graphs have connected $f$-factors, providing a polynomial-time algorithm to find such factors based on diameter considerations, extending understanding of the problem's complexity.

Contribution

It introduces a diameter-based characterization for connected $f$-factors in graphs with large minimum degree, and presents a polynomial-time algorithm for their detection.

Findings

Graphs with a connected $f$-factor and a 2-component $f$-factor have a diameter at least 3.

The algorithm combines Tutte's $f$-factor algorithm with diameter-based search.

The characterization applies when $f(v) extgreater rac{n}{2.5}$ for all vertices.

Abstract

It is well known that when $f(v)$ is a constant for each vertex $v$, the connected $f$-factor problem is NP-Complete. In this note we consider the case when $f(v) \geq \lceil \frac{n}{2.5}\rceil$ for each vertex $v$, where $n$ is the number of vertices. We present a diameter based characterization of graphs having a connected $f$-factor (for such $f$). We show that if a graph $G$ has a connected $f$-factor and an $f$-factor with 2 connected components, then it has a connected $f$-factor of diameter at least 3. This result yields a polynomial time algorithm which first executes the Tutte's $f$-factor algorithm, and if the output has 2 connected components, our algorithm searches for a connected $f$-factor of diameter at least 3.