On the structure of (pan, even hole)‐free graphs

A hole is a chordless cycle with at least four vertices. A pan is a graph that consists of a hole and a single vertex with precisely one neighbor on the hole. An even hole is a hole with an even number of vertices. We prove that a (pan, even hole)-free graph can be decomposed by clique cutsets into essentially unit circular-arc graphs. This structure theorem is the basis of our O(nm)-time certifying algorithm for recognizing (pan, even hole)-free graphs and for our O(n2.5+nm)-time algorithm to optimally color them. Using this structure theorem, we show that the tree-width of a (pan, even hole)-free graph is at most 1.5 times the clique number minus 1, and thus the chromatic number is at most 1.5 time the clique number.