Optimization Problems in Graph Theory pp 169-213 | Cite as

# Combinatorial and Graph-Theoretical Problems and Augmenting Technique

## Abstract

The notion of augmenting graphs generalizes Berge’s idea of augmenting chains, which was used by Edmonds in his celebrated so-called *Blossom Algorithm* for the maximum matching problem (Edmonds, Can J Math 17:449–467, 1965). This method then was developed for more general maximum independent set (MIS) problem, first for claw-free graphs (Minty, J Comb Theory Ser B 28(3):284–304, 1980; Sbihi, Discret Math 29(1):53–76, 1980). Then the method was used extensively for various special cases, for example, *S*_{1,1,2} (or fork)-free graphs (Alekseev, Discret Anal Oper Res Ser 1:3–19, 1999), subclasses of *P*_{5} free graphs (Boliac and Lozin, Discret Appl Math 131(3):567–575, 2003; Gerber et al., Discret Appl Math 132(1–3):109–119, 2004; Mosca, Discret Appl Math 132(1–3):175–183, 2004; Lozin and Mosca, Inf Process Lett 109(6):319–324, 2009), *P*_{6}-free graphs (Mosca, Discuss Math Graph Theory 32(3):387–401, 2012), *S*_{1,2,l}-free graphs (Hertz et al., Inf Process Lett 86(3):311–316, 2003; Hertz and Lozin, The maximum independent set problem and augmenting graphs. In: Graph theory and combinatorial optimization. Springer Science and Business Media, New York, pp. 69–99, 2005), *S*_{1,2,5}-free graphs (Lozin and Milanič, Discret Appl Math 156(13):2517–2529, 2008), and for *S*_{1,1,3}-free graphs (Dabrowski et al., Graphs Comb 32(4):1339–1352, 2016). In this paper, we will extend the method for some more general graph classes. Our objective is combining these approaches to apply this technique to develop polynomial time algorithms for the MIS problem in some special subclasses of *S*_{2,2,5}-free graphs, extending in this way different known results. Moreover, we also consider the augmenting technique for some other combinatorial graph-theoretical problems, for example maximum induced matching, maximum multi-partite induced subgraphs, maximum dissociative set, etc.

## Notes

### Acknowledgements

This research is supported by National Foundation for Science and Technology Development (NAFOSTED) of Vietnam, project code: 101.99-2016.20. We would like to express our special thanks to an anonymous reviewer for his/her very useful suggestions and comments.

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