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

The world of hereditary graph classes viewed through Truemper configurations

“…Note that chordless graphs were first studied in the 1960s by Dirac [66] and Plummer [104]. A description of their work can also be found in [3]. …”

“…Motivated by trying to understand the structure of wheel-free graphs, whose recognition remains an open problem, Aboulker, Radovanović, Trotignon and Vušković studied in [3] a subclass of wheel-free graphs known as propeller-free graphs. A propeller is a a graph that consists of a cycle C and a node x that has at least two neighbors on C. Let C 0 be the class of graphs that have no node that has at least two neighbors of degree at least 3, C 1 the class of graphs that have no propeller as a subgraph, and C 2 the class of propeller-free graphs.…”

“…First let us point out that by considering a longest path it is easy to show that graphs in C 2 must always have a node of degree at most 2, and hence they are 3-colorable, see [3]. Observe that since a clique on 4 nodes is a propeller finding the size of a largest clique in a propeller-free graph can easily be done in polynomial time.…”

“…The following decomposition theorems are given in [3], and used to obtain an O(nm) recognition algorithm for class C 1 , and an O(n 2 m 2 ) recognition algorithm for class C 2 .…”

“…is an edge). An S 2 -cutset is proper if nodes of G \ {a, b} can be partitioned into sets X and Y so that no node of X is adjacent to a node of Y , and Furthermore, it is shown in [3] that propeller-free graphs admit an extreme decomposition, which is used to prove that 2-connected propeller-free graphs must always have an edge whose endnodes are of degree 2. This implies that propellerfree graphs can also be edge-colored in polynomial time.…”