Chapter

Automata, Languages and Programming

Volume 5125 of the series Lecture Notes in Computer Science pp 587-596

Understanding the Complexity of Induced Subgraph Isomorphisms

  • Yijia ChenAffiliated withBASICS, Department of Computer Science, Shanghai Jiaotong University
  • , Marc ThurleyAffiliated withInstitut für Informatik, Humboldt-Universität zu Berlin
  • , Mark WeyerAffiliated withInstitut für Informatik, Humboldt-Universität zu Berlin

Abstract

We study left-hand side restrictions of the induced subgraph isomorphism problem: Fixing a class https://static-content.springer.com/image/chp%3A10.1007%2F978-3-540-70575-8_48/MediaObjects/978-3-540-70575-8_48_IEq1_HTML.png , for given graphs G https://static-content.springer.com/image/chp%3A10.1007%2F978-3-540-70575-8_48/MediaObjects/978-3-540-70575-8_48_IEq2_HTML.png and arbitrary H we ask for induced subgraphs of H isomorphic to G.

For the homomorphism problem this kind of restriction has been studied by Grohe and Dalmau, Kolaitis and Vardi for the decision problem and by Dalmau and Jonsson for its counting variant.

We give a dichotomy result for both variants of the induced subgraph isomorphism problem. Under some assumption from parameterized complexity theory, these problems are solvable in polynomial time if and only if https://static-content.springer.com/image/chp%3A10.1007%2F978-3-540-70575-8_48/MediaObjects/978-3-540-70575-8_48_IEq3_HTML.png contains no arbitrarily large graphs.

All classifications are given by means of parameterized complexity. The results are presented for arbitrary structures of bounded arity which implies, for example, analogous results for directed graphs.

Furthermore, we show that no such dichotomy is possible in the sense of classical complexity. That is, if https://static-content.springer.com/image/chp%3A10.1007%2F978-3-540-70575-8_48/MediaObjects/978-3-540-70575-8_48_IEq4_HTML.png there are classes https://static-content.springer.com/image/chp%3A10.1007%2F978-3-540-70575-8_48/MediaObjects/978-3-540-70575-8_48_IEq5_HTML.png such that the induced subgraph isomorphism problem on https://static-content.springer.com/image/chp%3A10.1007%2F978-3-540-70575-8_48/MediaObjects/978-3-540-70575-8_48_IEq6_HTML.png is neither in https://static-content.springer.com/image/chp%3A10.1007%2F978-3-540-70575-8_48/MediaObjects/978-3-540-70575-8_48_IEq7_HTML.png nor https://static-content.springer.com/image/chp%3A10.1007%2F978-3-540-70575-8_48/MediaObjects/978-3-540-70575-8_48_IEq8_HTML.png -complete. This argument may be of independent interest, because it is applicable to various parameterized problems.