Convex-Round and Concave-Round Graphs

We introduce two new classes of graphs which we call convex-round, respectively concave-round graphs. Convex-round (concave-round) graphs are those graphs whose vertices can be circularly enumerated so that the (closed) neighborhood of each vertex forms an interval in the enumeration. Hence the two classes transform into each other by taking complements. We show that both classes of graphs have nice structural properties. We observe that the class of concave-round graphs properly contains the class of proper circular arc graphs and, by a result of Tucker [ Pacific J. Math., 39 (1971), pp. 535--545], is properly contained in the class of general circular arc graphs. We point out that convex-round and concave-round graphs can be recognized in O(n+m) time (here n denotes the number of vertices and m the number of edges of the graph in question). We show that the chromatic number of a graph which is convex-round (concave-round) can be found in time O(n+m) (O(n2)). We describe optimal O(n+m) time algorithms for finding a maximum clique, a maximum matching, and a Hamiltonian cycle (if one exists) for the class of convex-round graphs. Finally, we pose a number of open problems and conjectures concerning the structure and algorithmic properties of the two new classes and a related third class of graphs.