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A colouring of a graph G = (V, E) is a mapping c : V → {1, 2,. . .} such that c(u) = c(v) if uv ∈ E; if |c(V)| ≤ k then c is a k-colouring. The Colouring problem is that of testing whether a given graph has a k-colouring for some given integer k. If a graph contains no induced subgraph isomorphic to any graph in some family H, then it is called H-free. The… (More)

If a graph has no induced subgraph isomorphic to any graph in a finite family The class of H-free graphs has bounded clique-width if and only if H is an induced subgraph of the 4-vertex path P4. We study the (un)boundedness of the clique-width of graph classes defined by two forbidden induced sub-graphs H1 and H2. Prior to our study it was not known whether… (More)

Let G be a bipartite graph, and let H be a bipartite graph with a fixed bipartition (BH , WH). We consider three different, natural ways of forbidding H as an induced subgraph in G.

Given two graphs H1 and H2, a graph G is (H1, H2)-free if it contains no subgraph isomorphic to H1 or H2. We continue a recent study into the clique-width of (H1, H2)-free graphs and present three new classes of (H1, H2)-free graphs that have bounded clique-width. We also show the implications of our results for the computational complexity of the Colouring… (More)

A graph is H-free if it has no induced subgraph isomorphic to H. Brandstädt, Engelfriet, Le and Lozin proved that the class of chordal graphs with independence number at most 3 has unbounded clique-width. Brandstädt, Le and Mosca erroneously claimed that the gem and the co-gem are the only two 1-vertex P4-extensions H for which the class of H-free chordal… (More)

The vertex colouring problem is known to be NP-complete in the class of triangle-free graphs. Moreover, it remains NP-complete even if we additionally exclude a graph F which is not a forest. We study the computational complexity of the problem in (K 3 , F)-free graphs with F being a forest. From known results it follows that for any forest F on 5 vertices,… (More)

The maximum independent set problem is known to be NP-hard for graphs in general, but is solvable in polynomial time for graphs in many special classes. It is also known that the problem is generally intractable from a parameterized point of view. A simple Ramsey argument implies the fixed parameter tractability of the maximum independent set problem in… (More)

The maximum induced matching problem is known to be APX-hard in the class of bipartite graphs. Moreover, the problem is also intractable in this class from a parameterized point of view, i.e. it is W[1]-hard. In this paper, we reveal several classes of bipartite (and more general) graphs for which the problem admits fixed-parameter tractable algorithms. We… (More)

The maximum independent set problem is NP-complete for graphs in general, but becomes solvable in polynomial time when restricted to graphs in many special classes. The problem is also intractable from a parameterized point of view. However, very little is known about parameterized complexity of the problem in restricted graph classes. In the present paper,… (More)