The global structure of locally chordal graphs

Abstract

A graph is locally chordal if each of its small-radius balls is chordal. In an earlier work [AKK25], the authors and Kobler proved that locally chordal graphs can be characterized by having chordal local covers, by forbidding short cycles and wheels as induced subgraphs, and by the property that each of their minimal local separators is a clique. In this paper, we address the global structure of locally chordal graphs. The global structure of chordal graphs is given by the following characterizations: a graph is chordal if and only if it is the intersection graph of subtrees of a tree, if and only if it admits a tree-decomposition into cliques. We prove a local analog of this characterization, which essentially says that a graph is locally chordal if and only if it is the intersection graph of special subtrees of a high-girth graph, if and only if it admits a special graph-decomposition over a high-girth graph into cliques. We also prove that these global representations of locally chordal graphs can be efficiently computed. This paper has two major contributions. The first is to exhibit for locally chordal graphs an ideal "local to global" analysis: given a graph class defined by restricted local structure, we fully describe the global structure of graphs in the class. The second is to develop the theory of graph-decompositions. Much of the work in this paper is devoted to properties of graph-decompositions that represent the global structure of graphs. This theory will be useful to find global decompositions for graph classes beyond locally chordal graphs.