Ordering without forbidden patterns
Pavol Hell, Bojan Mohar, Arash Rafiey

TL;DR
This paper proves that determining if a graph admits an F-free ordering for small patterns can be done in polynomial time, providing a unified algorithm and forbidden structure characterization for these classes.
Contribution
The authors present a single polynomial-time algorithm for all sets F of patterns with up to three vertices, completing the classification of these problems.
Findings
Polynomial-time algorithm for ORD_3 with small patterns
Forbidden structure characterization for graph classes
NP-completeness examples for larger patterns
Abstract
Let F be a set of ordered patterns, i.e., graphs whose vertices are linearly ordered. An F-free ordering of the vertices of a graph H is a linear ordering of V(H) such that none of patterns in F occurs as an induced ordered subgraph. We denote by ORD(F) the decision problem asking whether an input graph admits an F-free ordering; we also use ORD(F) to denote the class of graphs that do admit an F-free ordering. It was observed by Damaschke (and others) that many natural graph classes can be described as ORD(F) for sets F of small patterns (with three or four vertices). Damaschke also noted that for many sets F consisting of patterns with three vertices, ORD(F) is polynomial-time solvable by known algorithms or their simple modifications. We complete the picture by proving that all these problems can be solved in polynomial time. In fact, we provide a single master algorithm, i.e., we…
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Taxonomy
TopicsAdvanced Graph Theory Research · semigroups and automata theory · Complexity and Algorithms in Graphs
