Vadim V. Lozin

Recent developments on graphs of bounded clique-width

Whether the clique-width of graphs in a certain class of graphs is bounded or not, is an important question from an algorithmic point of view, as many problems that are NP-hard in general admit polynomial-time solutions when restricted to graphs of bounded clique-width. Over the last few years, many classes of graphs have been shown to have bounded clique-width. For many others, this parameter has been proved to be unbounded. This paper provides a survey of recent results addressing this problem.

On maximum induced matchings in bipartite graphs

Abstract The problem of finding a maximum induced matching is known to be NP-hard in general bipartite graphs. We strengthen this result by reducing the problem to some special classes of bipartite graphs such as bipartite graphs with maximum degree 3 or C 4 -free bipartite graphs. On the other hand, we describe a new polynomially solvable case for the problem in bipartite graphs which deals with a generalization of bi-complement reducible graphs.

On the Band-, Tree-, and Clique-Width of Graphs with Bounded Vertex Degree

The band-, tree-, and clique-width are of primary importance in algorithmic graph theory due to the fact that many problems that are NP-hard for general graphs can be solved in polynomial time when restricted to graphs where one of these parameters is bounded. It is known that for any fixed

Some results on graphs without long induced paths

\Delta \geq 3

Independent sets in extensions of 2K 2 -free graphs

, all three parameters are unbounded for graphs with vertex degree at most

NP-hard graph problems and boundary classes of graphs

\Delta

Clique-Width for 4-Vertex Forbidden Subgraphs

. In this paper, we distinguish representative subclasses of graphs with bounded vertex degree that have bounded band-, tree-, or clique-width. Our proofs are constructive and lead to efficient algorithms for a variety of NP-hard graph problems when restricted to those classes.

On the Clique-Width of Graphs in Hereditary Classes

Many NP-hard graph problems remain difficult on Pk-free graphs for certain values of k. Our goal is to distinguish subclasses of Pk-free graphs where several important graph problems can be solved in polynomial time. In particular, we show that the independent set problem is polynomial-time solvable in the class of (Pk, K1,n)-free graphs for any positive integers k and n, thereby generalizing several known results.

Stability in P5- and banner-free graphs

The class of 2K2-free graphs includes several interesting subclasses such as split, pseudo-split, threshold graphs, complements to chordal, interval or trivially perfect graphs. The fundamental property of 2K2-free graphs is that they contain polynomially many maximal independent sets. As a consequence, several important problems that are NP-hard in general graphs, such as 3- colorability, maximum weight independent set (WIS), minimum weight independent dominating set (WID), become polynomial-time solvable when restricted to the class of 2K2-free graphs. In the present paper, we extend 2K2-free graphs to larger classes with polynomial-time solvable WIS or WID. In particular, we show that WIS can be solved in polynomial time for (K2 + K1,3)- free graphs and WID for (K2 + K1,2)-free graphs. The latter result is in contrast with the fact that independent domination is NP-hard in the class of 2K1,2-free graphs, which has been recently proven by Zverovich.

Independent Sets of Maximum Weight in Apple-Free Graphs

Any graph problem, which is NP-hard in general graphs, becomes polynomial-time solvable when restricted to graphs in special classes. When does a difficult problem become easy? To answer this question we study the notion of boundary classes. In the present paper we define this notion in its most general form, describe several approaches to identify boundary classes and apply them to various graph problems.