Combinatorics and algorithms for quasi-chain graphs

TL;DR

This paper explores the complexity of quasi-chain graphs, showing they are as complex as permutations and NP-complete for induced subgraph isomorphism, but also provides a decomposition theorem enabling efficient algorithms.

Contribution

It establishes the complexity of quasi-chain graphs and introduces a decomposition theorem for efficient algorithms and implicit representations.

Findings

Induced subgraph isomorphism is NP-complete for quasi-chain graphs.

A bijection between permutations and a subclass of quasi-chain graphs is constructed.

A decomposition theorem enables efficient algorithms for certain problems.

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.