# On the Complexity Landscape of Connected f -Factor Problems

## TL;DR

This paper explores the computational complexity of connected f-factor problems in graphs, revealing new thresholds where the problem shifts from polynomial-time solvable to NP-intermediate, depending on the lower bounds of the function f.

## Contribution

It extends previous work by establishing new complexity results for connected f-factor problems based on restrictions on the function f.

## Key findings

- Connected f-factor is NP-complete in general.
- When f(v) >= n/(log n)^c, the problem is quasi-polynomial or randomized polynomial-time solvable for c<=1.
- For c>1, the problem is NP-intermediate.

## Abstract

Let G be an undirected simple graph having n vertices and let f be a function defined to be f:V(G) -> {0,..., n-1}. An f-factor of G is a spanning subgraph H such that degree of a vertex v in H is f(v) for every vertex v in V(G). The subgraph H is called a connected f-factor if, in addition, H is connected. A classical result of Tutte(1954) is the polynomial time algorithm to check whether a given graph has a specified f-factor. However, checking for the presence of a connected f-factor is easily seen to generalize HAMILTONIAN CYCLE and hence is NP-complete. In fact, the CONNECTED f-FACTOR problem remains NP-complete even when we restrict f(v) to be at least n^e for each vertex v and 0<e<1; on the other side of the spectrum of nontrivial lower bounds on f, the problem is known to be polynomial time solvable when f(v) is at least n/3 for every vertex v. In this paper, we extend this line of work and obtain new complexity results based on restrictions on the function f. In particular, we show that when f(v) is restricted to be at least n/(log n)^c , the problem can be solved in quasi-polynomial time in general and in randomized polynomial time if c<=1. Furthermore, we show that when c>1, the problem is NP-intermediate.