Augmenting chains in graphs without a skew star

Abstract

The augmenting chain technique has been applied to solve the maximum stable set problem in the class of line graphs (which coincides with the maximum matching problem) and then has been extended to the class of claw-free graphs. In the present paper, we propose a further generalization of this approach. Specifically, we show how to find an augmenting chain in graphs containing no skew star, i.e. a tree with exactly three vertices of degree 1 of distances 1, 2, 3 from the only vertex of degree 3. As a corollary, we prove that the maximum stable set problem is polynomially solvable in a class that strictly contains claw-free graphs, improving several existing results.

From matchings to independent sets

“…Later, in 1980, this solution was extended to claw-free graphs by Minty and Sbihi. Much later, in 2006, a polynomial-time algorithm for detecting augmenting paths was developed for S 1,2,3 -free graphs [19]. This class provides a vast generalization of clawfree graphs, but the algorithm for detecting augmenting paths does not solve the maximum independent set problem for S 1,2,3 -free graphs, because this class contains other types of augmenting graphs.…”

“…The second step is developing algorithms for detecting all types of augmenting graphs in these classes. Clearly, both of them contain augmenting paths and, fortunately, in both of them augmenting paths can be detected in polynomial time, since augmenting paths can be found efficiently in S 1,2,3 -free graphs [19]. Determining the complexity of finding augmenting paths in S i,j,k -free graphs for larger values of i, j, k is another open questions that would be interesting to investigate.…”

Combinatorics and Algorithms for Augmenting Graphs

“…In this section we deal with the problem of finding augmenting graphs in (S 1,1,3 , K p,p )-free According to Theorem 8, this problem consists of two main subproblems: finding augmenting paths and finding extensions of simple trees. The first of these was solved in [5] even for more general graphs, namely for S 1,2,3 -free graphs. In Lemma 9 we solve the second subproblem.…”

“…Proof. Let G be an (S 1,1,3 , K p,p )-free graph and S an arbitrary independent set in G. If G contains an augmenting path for S, such a path can be found by an algorithm proposed in [5], which works in polynomial time for any graph containing no induced S 1,2,3 .…”