Nicholas Korpelainen

Boundary properties of graphs for algorithmic graph problems

The notion of a boundary graph property was recently introduced as a relaxation of that of a minimal property and was applied to several problems of both algorithmic and combinatorial nature. In the present paper, we first survey recent results related to this notion and then apply it to two algorithmic graph problems: Hamiltonian cycle and vertexk-colorability. In particular, we discover the first two boundary classes for the Hamiltonian cycle problem and prove that for any k>3 there is a continuum of boundary classes for vertexk-colorability.

A polynomial-time algorithm for the dominating induced matching problem in the class of convex graphs

The dominating induced matching problem is the problem of determining whether a graph has an induced matching that dominates every edge of the graph. This is known to be NP-complete in general. We develop a polynomial-time algorithm to solve the problem for convex graphs.

Split Permutation Graphs

The class of split permutation graphs is the intersection of two important classes, the split graphs and permutation graphs. It also contains an important subclass, the threshold graphs. The class of threshold graphs enjoys many nice properties. In particular, these graphs have bounded clique-width and they are well-quasi-ordered by the induced subgraph relation. It is known that neither of these two properties is extendable to split graphs or to permutation graphs. In the present paper, we study the question of extendability of these two properties to split permutation graphs. We answer this question negatively with respect to both properties. Moreover, we conjecture that with respect to both of them the split permutation graphs constitute a critical class.

Bipartite induced subgraphs and well-quasi-ordering

We study bipartite graphs partially ordered by the induced subgraph relation. Our goal is to distinguish classes of bipartite graphs that are or are not well-quasi-ordered (wqo) by this relation. Answering an open question from [J Graph Theory 16 (1992), 489–502], we prove that P7-free bipartite graphs are not wqo. On the other hand, we show that P6-free bipartite graphs are wqo. We also obtain some partial results on subclasses of bipartite graphs defined by forbidding more than one induced subgraph. Copyright

Boundary Properties of Well-Quasi-Ordered Sets of Graphs

Let

Dominating induced matchings in graphs without a skew star

{\cal Y}_k

Bipartite Graphs of Large Clique-Width

be the family of hereditary classes of graphs defined by k forbidden induced subgraphs. In Korpelainen and Lozin (Discrete Math 311:1813–1822, 2011), it was shown that

Linear clique-width for hereditary classes of cographs

{\cal Y}_2

Infinitely many minimal classes of graphs of unbounded clique-width

contains only finitely many minimal classes that are not well-quasi-ordered (wqo) by the induced subgraph relation. This implies, in particular, that the problem of deciding whether a class from

Hamiltonian cycles in subcubic graphs: what makes the problem difficult

{\cal Y}_2