Geodesic
On Almost Well-Covered Graphs of Girth at Least 6
Discrete mathematics & theoretical computer science, Tome 20 (2018) no. 2.

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We consider a relaxation of the concept of well-covered graphs, which are graphs with all maximal independent sets of the same size. The extent to which a graph fails to be well-covered can be measured by its independence gap, defined as the difference between the maximum and minimum sizes of a maximal independent set in $G$. While the well-covered graphs are exactly the graphs of independence gap zero, we investigate in this paper graphs of independence gap one, which we also call almost well-covered graphs. Previous works due to Finbow et al. (1994) and Barbosa et al. (2013) have implications for the structure of almost well-covered graphs of girth at least $k$ for $k\in \{7,8\}$. We focus on almost well-covered graphs of girth at least $6$. We show that every graph in this class has at most two vertices each of which is adjacent to exactly $2$ leaves. We give efficiently testable characterizations of almost well-covered graphs of girth at least $6$ having exactly one or exactly two such vertices. Building on these results, we develop a polynomial-time recognition algorithm of almost well-covered $\{C_3,C_4,C_5,C_7\}$-free graphs. 
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     author = {Ekim, T{\i}naz and G\"oz\"upek, Didem and Hujdurovi\'c, Ademir and Milani\v{c}, Martin},
     title = {On {Almost} {Well-Covered} {Graphs} of {Girth} at {Least} 6},
     journal = {Discrete mathematics & theoretical computer science},
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     number = {2},
     year = {2018},
     doi = {10.23638/DMTCS-20-2-17},
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Ekim, Tınaz; Gözüpek, Didem; Hujdurović, Ademir; Milanič, Martin. On Almost Well-Covered Graphs of Girth at Least 6. Discrete mathematics & theoretical computer science, Tome 20 (2018) no. 2. doi : 10.23638/DMTCS-20-2-17. http://geodesic.mathdoc.fr/articles/10.23638/DMTCS-20-2-17/

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