# Algorithmic Complexity of Weakly Semiregular Partitioning and the Representation Number

## TL;DR

This paper investigates the algorithmic complexity of partitioning graphs into weakly semiregular subgraphs and explores the computational difficulty of determining the representation number of graphs, providing new algorithms and complexity results.

## Contribution

It presents a polynomial-time algorithm for checking if a tree's weakly semiregular number is two and proves NP-completeness for bipartite graphs with degree constraints; it also confirms the conjecture on the complexity of computing the representation number.

## Key findings

- Polynomial-time algorithm for trees with weakly semiregular number two
- NP-completeness for bipartite graphs with degree constraints
- No efficient approximation for representation number unless P=NP

## Abstract

A graph $G$ is {\it weakly semiregular} if there are two numbers $a,b$, such that the degree of every vertex is $a$ or $b$. The {\it weakly semiregular number} of a graph $G$, denoted by $wr(G)$, is the minimum number of subsets into which the edge set of $G$ can be partitioned so that the subgraph induced by each subset is a weakly semiregular graph. We present a polynomial time algorithm to determine whether the weakly semiregular number of a given tree is two. On the other hand, we show that determining whether $ wr(G) = 2 $ for a given bipartite graph $ G $ with at most three numbers in its degree set is {\bf NP}-complete. Among other results, for every tree $T$, we show that $wr(T)\leq 2\log_2 \Delta(T) + \mathcal{O}(1)$, where $\Delta(T)$ denotes the maximum degree of $T$. In the second part of the work, we consider the representation number. A graph $G$ has a {\it representation modulo $r$} if there exists an injective map $\ell: V (G) \rightarrow \mathbb{Z}_r$ such that vertices $v$ and $u$ are adjacent if and only if $|\ell(u) -\ell(v)|$ is relatively prime to $r$. The {\it representation number}, denoted by $rep(G)$, is the smallest $r$ such that $G$ has a representation modulo $r$. Narayan and Urick conjectured that the determination of $rep (G)$ for an arbitrary graph $G$ is a difficult problem \cite{narayan2007representations}. In this work, we confirm this conjecture and show that if $\mathbf{NP\neq P}$, then for any $\epsilon >0$, there is no polynomial time $(1-\epsilon)\frac{n}{2}$-approximation algorithm for the computation of representation number of regular graphs with $n$ vertices.