Decompositions of complete multigraphs into stars of varying sizes

TL;DR

This paper investigates the complexity of decomposing complete multigraphs into stars of specified sizes, establishing NP-completeness in general but tractability under certain size constraints, and extends related graph decomposition theorems.

Contribution

It proves NP-completeness for the general problem and identifies the size bound where the problem becomes tractable, also providing characterizations and generalizations of existing theorems.

Findings

NP-complete for general multigraph star decompositions

Tractable when maximum star size is below a specific bound

Extended Landau's theorem to new graph decomposition contexts

Abstract

In 1979 Tarsi showed that an edge decomposition of a complete multigraph into stars of size $m$ exists whenever some obvious necessary conditions hold. In 1992 Lonc gave necessary and sufficient conditions for the existence of an edge decomposition of a (simple) complete graph into stars of sizes $m_1,\ldots,m_t$. We show that the general problem of when a complete multigraph admits a decomposition into stars of sizes $m_1,\ldots,m_t$ is $\mathsf{NP}$-complete, but that it becomes tractable if we place a strong enough upper bound on $\max(m_1,\ldots,m_t)$. We determine the upper bound at which this transition occurs. Along the way we also give a characterisation of when an arbitrary multigraph can be decomposed into stars of sizes $m_1,\ldots,m_t$ with specified centres, and a generalisation of Landau's theorem on tournaments.