Geodeticity of the contour of chordal graphs Original Research Article

A vertex image is a boundary vertex of a connected graph G if there exists a vertex u such that no neighbor of image is further away from u than image. Moreover, if no vertex in the whole graph image is further away from u than image, then image is called an eccentric vertex of G. A vertex image belongs to the contour of G if no neighbor of image has an eccentricity greater than the eccentricity of image. Furthermore, if no vertex in the whole graph image has an eccentricity greater than the eccentricity of image, then image is called a peripheral vertex of G. This paper is devoted to study these kinds of vertices for the family of chordal graphs. Our main contributions are, firstly, obtaining a realization theorem involving the cardinalities of the periphery, the contour, the eccentric subgraph and the boundary, and secondly, proving both that the contour of every chordal graph is geodetic and that this statement is not true for every perfect graph.