Decomposing a graph into subgraphs with small components

TL;DR

This paper studies the computational complexity of decomposing graphs into subgraphs with small components, proving NP-completeness in various cases and providing algorithms for optimal decompositions and related problems.

Contribution

It establishes NP-completeness of the $(k,c)$-Decomposition problem in bipartite graphs and trees, and offers fixed-parameter and approximation algorithms for related graph decomposition tasks.

Findings

NP-complete in bipartite graphs for fixed k and c

NP-complete in trees when k and c are part of the input

FPT algorithm for fixed c and approximation algorithms for optimal decompositions

Abstract

The component size of a graph is the maximum number of edges in any connected component of the graph. Given a graph $G$ and two integers $k$ and $c$, $(k,c)$-Decomposition is the problem of deciding whether $G$ admits an edge partition into $k$ subgraphs with component size at most $c$. We prove that for any fixed $k \ge 2$ and $c \ge 2$, $(k,c)$-Decomposition is NP-complete in bipartite graphs. Also, when both $k$ and $c$ are part of the input, $(k,c)$-Decomposition is NP-complete even in trees. Moreover, $(k,c)$-Decomposition in trees is W[1]-hard with parameter $k$, and is FPT with parameter $c$. In addition, we present approximation algorithms for decomposing a tree either into the minimum number of subgraphs with component size at most $c$, or into $k$ subgraphs minimizing the maximum component size. En route to these results, we also obtain a fixed-parameter algorithm for Bin Packing with the bin capacity as parameter.