Boundary-type sets in maximal outerplanar graphs

Abstract

The periphery Per(G) of a graph G is the set of vertices of maximum eccentricity. A vertex v belongs to the contour Ct(G) of G if no neighbor of v has an eccentricity greater than the eccentricity of v. The extreme set Ext(G) of G is the set of all its simplicial (also called extreme) vertices; an extreme vertex is such that its neighborhood induces a complete graph. The eccentricity Ecc(G) of a graph G is the set of all its eccentric vertices, i.e. vertices that are antipodal to some other vertex in G. A vertex v is a boundary vertex if there is another vertex u in G such that no u-v geodesic can be extended at v to a longer geodesic. The boundary∂(G) of G is the set of all its boundary vertices.For the family of maximal outerplanar graphs, we provide a characterization of ∂(G) and Ext(G) in terms of vertex degrees. We characterize those graphs that are induced by Per(G) and Ct(G). We show that, unlike for trees, all relationships between boundary-type sets in maximal outerplanar graphs are as rich as in general graphs and we construct a single maximal outerplanar graph showing the sharpness of all those inclusions.