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

TL;DR

This paper provides structural characterizations and polynomial algorithms for recognizing and coloring graphs that exclude a specific cycle with a node having multiple neighbors on it, extending known results on minimally 2-connected graphs.

Contribution

It introduces new structural characterizations and polynomial algorithms for classes of graphs avoiding a cycle with a node of multiple neighbors, generalizing previous results.

Findings

Structural characterizations of the graph classes

Polynomial time recognition algorithms

Polynomial algorithms for vertex and edge coloring

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, as well as polynomial time algorithms for vertex-coloring and edge-coloring.