Graphs of low chordality

The chordality of an undirected graph G, which is not acyclic, is defined as the length of the longest induced cycle in it. The chordality of an acyclic graph is defined to be 0. We use Cl (l ≥ 3) to denote a cycle of length l. An induced cycle is called a hole. A hole is an odd hole if its length is odd and is an even hole otherwise. Odd-chordality of a graph is the length of the longest odd hole in it. Even-chordality of a graph is the length of the longest even hole in it. In the present paper we identify several classes of graphs of bounded chordality. Our motivation is due to some recent interesting results connecting chordality with other structural aspects of graphs. We list some of them below.