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, and Whitehead implies a polynomial-time recognition algorithm for the class of almost well-covered graphs of girth at least 8. We focus on the structure of almost well-covered graphs of girth at least 6. We give a complete structural characterization of a subclass of the class of almost well-covered graphs of girth at least 6, a polynomially testable characterization of another one, and a polynomial-time recognition algorithm of almost wellcovered {C3, C4, C5, C7}-free graphs.