Interval graphs: canonical representation in logspace
ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming
From polynomial time queries to graph structure theory
Communications of the ACM
Fixed-point definability and polynomial time on chordal graphs and line graphs
Fields of logic and computation
Interval Graphs: Canonical Representations in Logspace
SIAM Journal on Computing
Fixed-point definability and polynomial time on graphs with excluded minors
Journal of the ACM (JACM)
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We prove a characterization of all polynomialtime computable queries on the class of interval graphs by sentences of fixed-point logic with counting. More precisely, it is shown that on the class of unordered interval graphs, any query is polynomial-time computable if and only if it is definable in fixed-point logic with counting. This result is one of the first establishing the capturing of polynomial time on a graph class which is defined by forbidden induced subgraphs. For this, we define a canonical form of interval graphs using a type of modular decomposition, which is different from the method of tree decomposition that is used in most known capturing results for other graph classes, specifically those defined by forbidden minors. The method might also be of independent interest for its conceptual simplicity. Furthermore, it is shown that fixed-point logic with counting is not expressive enough to capture polynomial time on the classes of chordal graphs or incomparability graphs.