Decomposing Cubic Graphs into Connected Subgraphs of Size Three

TL;DR

This paper investigates the complexity of partitioning cubic graphs into connected subgraphs of size three, identifying which cases are polynomial-time solvable and which are NP-complete, with some characterizations of decomposable graphs.

Contribution

It classifies the computational complexity of edge partitioning problems in cubic graphs for various subsets of connected size-three subgraphs, providing new characterizations.

Findings

Identifies polynomial and NP-complete cases for cubic graph decompositions.

Provides graph-theoretic characterizations of decomposable cubic graphs.

Complements existing NP-completeness results with specific cases.

Abstract

Let $S=\{K_{1,3},K_3,P_4\}$ be the set of connected graphs of size 3. We study the problem of partitioning the edge set of a graph $G$ into graphs taken from any non-empty $S'\subseteq S$. The problem is known to be NP-complete for any possible choice of $S'$ in general graphs. In this paper, we assume that the input graph is cubic, and study the computational complexity of the problem of partitioning its edge set for any choice of $S'$. We identify all polynomial and NP-complete problems in that setting, and give graph-theoretic characterisations of $S'$-decomposable cubic graphs in some cases.