On finding augmenting graphs

• Published in: Discrete Applied Mathematics Vol. 156; no. 13; pp. 2517 - 2529
• Main Authors: Lozin, Vadim V., Milanič, Martin
• Format: Journal Article Conference Proceeding
• Language: English
• Published: Lausanne Elsevier B.V 06.07.2008; Amsterdam Elsevier; New York, NY
• Subjects: Algorithmics. Computability. Computer arithmetics; Applied sciences; Augmenting graph; Combinatorics; Combinatorics. Ordered structures; Computer science; control theory; systems; Exact sciences and technology; Independent set; Information retrieval. Graph; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Partial differential equations, initial value problems and time-dependant initial-boundary value problems; Polynomial-time algorithm; Sciences and techniques of general use; Theoretical computing; Independent set; Augmenting graph; Polynomial-time algorithm; Computer theory; Implementation; Optimization; Polynomial time; Maximum; Line graph; Graph chain; NP hard problem; Method of lines; Graph method; Combinatorics; Graph matching
• ISSN: 0166-218X; 1872-6771
• DOI: 10.1016/j.dam.2008.03.008

Method of augmenting graphs is a general approach to solve the maximum independent set problem. As the problem is generally NP-hard, no polynomial time algorithms are available to implement the method. However, when restricted to particular classes of graphs, the approach may lead to efficient solutions. A famous example of this type is the maximum matching algorithm: it finds a maximum matching in a graph G , which is equivalent to finding a maximum independent set in the line graph of G . In the particular case of line graphs, the method reduces to finding augmenting (alternating) chains. Recent investigations of more general classes of graphs revealed many more types of augmenting graphs. In the present paper we study the problem of finding augmenting graphs different from chains. To simplify this problem, we introduce the notion of a redundant set. This allows us to reduce the problem to finding some basic augmenting graphs. As a result, we obtain a polynomial time solution to the maximum independent set problem in a class of graphs which extends several previously studied classes including the line graphs.