Intersection of chordal graphs and some related partition problems

Abstract

The chordality of a graph is the minimum number of chordal graphs whose intersection is the graph. A result of Yannakakis’ from 1982 can be used to infer that for every fixed $k\ge 3$, deciding whether the chordality of a graph is at most $k$ is NP-complete. We consider the problem of deciding whether the chordality of a graph is 2, or equivalently, deciding whether a given graph is the intersection of two chordal graphs. We prove that the problem is equivalent to a partition problem when one of the chordal graphs is a split graph and the other meets certain conditions. Using this we derive complexity results for a variety of problems, including deciding if a graph is the intersection of $k$ split graphs, which is in P for $k=2$ and NP-complete for $k\ge 3$.