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arXiv · 2104.04471

Combinatorics and algorithms for quasi-chain graphs

Bogdan Alecu·Aistis Atminas·Vadim Lozin·Dmitriy Malyshev
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Abstract

The class of quasi-chain graphs is an extension of the well-studied class of chain graphs. This latter class enjoys many nice and important properties, such as bounded clique-width, implicit representation, well-quasi-ordering by induced subgraphs, etc. The class of quasi-chain graphs is substantially more complex. In particular, this class is not well-quasi-ordered by induced subgraphs, and the clique-width is not bounded in it. In the present paper, we show that the universe of quasi-chain graphs is at least as complex as the universe of permutations by establishing a bijection between the class of all permutations and a subclass of quasi-chain graphs. This implies, in particular, that the induced subgraph isomorphism problem is NP-complete for quasi-chain graphs. On the other hand, we propose a decomposition theorem for quasi-chain graphs that implies an implicit representation for graphs in this class and efficient solutions for some algorithmic problems that are generally intractable.

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Bogdan Alecu, Aistis Atminas, Vadim Lozin, Dmitriy Malyshev. 2021-04-09. Combinatorics and algorithms for quasi-chain graphs. https://arxiv.org/abs/2104.04471

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