Graphs that do not contain a cycle with a node that has at least two neighbors on it

Abstract

We recall several known results about minimally 2-connected graphs and show that they all follow from a decomposition theorem. Starting from an analogy with critically 2-connected graphs, we give structural characterizations of the classes of graphs that do not contain as a subgraph and as an induced subgraph, a cycle with a node that has at least two neighbors on the cycle. From these characterizations we get polynomial time recognition algorithms for these classes and polynomial time algorithms for vertex-coloring and edge-coloring. Copyright © by SIAM.