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Complexity of generalized colourings of chordal graphs

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(Thesis) Ph.D.
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The generalized graph colouring problem (GCOL) for a fixed integer k, and fixed classes of graphs P_1,...,P_k (usually describing some common graph properties), is to decide, for a given graph G, whether the vertex set of G can be partitioned into sets V_1,..., V_k such that, for each i, the induced subgraph of G on V_i belongs to P_i. It can be seen that GCOL generalizes many natural colouring and partitioning problems on graphs. In this thesis, we focus on generalized colouring problems in chordal graphs. The structure of chordal graphs is known to allow solving many difficult combinatorial problems, such as the graph colouring, maximum clique and others, in polynomial, and in many cases in linear time. Our study of generalized colouring problems focuses on those problems in which the sets P_i are characterized by a single forbidden induced subgraph. We show, that for k=2, all such problems where the forbidden graphs have at most three vertices are polynomial time solvable in chordal graphs, whereas, it is known that almost all of them are NP-complete in general. On the other hand, we show infinite families of such problems which are NP-complete in chordal graphs. By combining a polynomial algorithm and an NP-completeness proof, we answer a question of Broersma, Fomin, Nesetril and Woeginger about the complexity of the so-called subcolouring problem on chordal graphs. Additionally, we explain, how some of these results generalize to particular subclasses of chordal graphs, and we show a complete forbidden subgraph characterization for the so-called monopolar partitions of chordal graphs. Finally, in the last part of the thesis, we focus on a different type of colouring problem -- injective colouring. We describe several algorithmic and (in-)approximability results for injective colourings in the class of chordal graphs and its subclasses. In the process, we correct a result of Agnarsson et al. on inapproximability of the chromatic number of the square of a split graph.
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