On Chordal and Perfect Plane Triangulations

We investigate a method of decomposing a plane near-triangulation G into a collection of induced component subgraphs which we call the W components of the graph. Each W component is essentially a plane near-triangulation with the property that the neighbourhood of every internal vertex induces a wheel. The problem of checking whether a plane near-triangulation G is chordal (or perfect) is shown to be transformable to the problem of checking whether its W components are chordal (or perfect). Using this decomposition method, we show that a plane near-triangulated graph is chordal if and only if it does not contain an internal vertex whose closed neighbourhood induces a wheel of at least five vertices. Though a simple local characterization for plane perfect near-triangulations is unlikely, we show that there exists a local characterization for perfect W components that does not contain any induced wheel of five vertices.