Sequences of radius $k$ for complete bipartite graphs

TL;DR

This paper investigates the length of sequences that cover all edges within distance k in complete bipartite graphs, providing tight estimates and proving NP-hardness for general graphs when k>1.

Contribution

It offers an asymptotically tight estimate for the sequence length in complete bipartite graphs and proves NP-hardness of computing this length for arbitrary graphs when k>1.

Findings

Asymptotically tight estimate for f_k(G) in complete bipartite graphs.

NP-hardness of determining f_k(G) for arbitrary graphs for k>1.

Lower bound valid for all bipartite graphs.

Abstract

A \emph{$k$-radius sequence} for a graph $G$ is a sequence of vertices of $G$ (typically with repetitions) such that for every edge $uv$ of $G$ vertices $u$ and $v$ appear at least once within distance $k$ in the sequence. The length of a shortest $k$-radius sequence for $G$ is denoted by $f_k(G)$. We give an asymptotically tight estimation on $f_k(G)$ for complete bipartite graphs {which matches a lower bound, valid for all bipartite graphs}. We also show that determining $f_k(G)$ for an arbitrary graph $G$ is NP-hard for every constant $k>1$.