Computational complexity of the Krausz dimension of graphs

Abstract

A Krausz partition of a graph G is a partition of the edges of G into complete subgraphs. The Krausz dimension of a graph G is the least number k such that G admits a Krausz partition in which each vertex belongs to at most k classes. The graphs with Krausz dimension at most 2 k are exactly the line graphs, and graphs of the Krausz dimension at most k are intersection graphs of k-uniform linear hypergraphs. This paper studies the computational complexity of the Krausz dimension problem. We show that deciding if Krausz dimension of a graph is at most 3 is NP-complete in general, but solvable in polynomial time for graphs of maximum. degree 4. We pay closer attention to chordal graphs, showing that deciding if Krausz dimension is at most 6 is NP-complete for chordal graphs in general, while the Krausz dimension of a chordal graph with bounded clique size can be determined in polynomial time. We also show that for any fixed k, it can be decided in polynomial time if an interval graph has Krausz dimension at most k.