A Comprehensive Review on a Class of Planar Well- Covered Graphs with Girth Four

A class of planar well-covered graphs with girth four

Almost Well-Covered Graphs Without Short Cycles

We study graphs in which the maximum and the minimum sizes of a maximal independent set differ by exactly one. We call these graphs almost well-covered, in analogy with the class of well-covered graphs, in which all maximal independent sets have the same size. A characterization of graphs of girth at least 8 having exactly two different sizes of maximal independent sets due to Finbow, Hartnell,...

Oriented coloring is NP-complete on a small class of graphs

A series of recent papers shows that it is NP-complete to decide whether an oriented graph admits a homomorphism to the tournament T4 on 4 vertices containing a 4-circuit, each time on a smaller graph class. We improve these results by showing that homomorphism to T4 is NP-complete for bipartite planar subcubic graphs of arbitrarily large fixed girth. We also show that push homomorphism is NP-c...

On universal graphs for planar oriented graphs of a given girth

The oriented chromatic number o(H) of an oriented graph H is defined to be the minimum order of an oriented graph H ' such that H has a homomorphism to H' . If each graph in a class ~ has a homomorphism to the same H' , then H ' is ~-universal. Let ~k denote the class of orientations of planar graphs with girth at least k. Clearly, ~3 ~ ~4 ~ ~5... We discuss the existence of ~k-universal graphs...

On well-covered graphs of odd girth 7 or greater

A maximum independent set of vertices in a graph is a set of pairwise nonadjacent vertices of largest cardinality α. Plummer [14] defined a graph to be well-covered, if every independent set is contained in a maximum independent set of G. One of the most challenging problems in this area, posed in the survey of Plummer [15], is to find a good characterization of well-covered graphs of girth 4. ...