Induced subgraph isomorphism problem

In complexity theory and graph theory, induced subgraph isomorphism is an NP-complete decision problem that involves finding a given graph as an induced subgraph of a larger graph.

Maximum lengths of snakes (L_s) and coils (L_c) in the snakes-in-the-box problem for dimensions n from 1 to 4

Problem statement

Formally, the problem takes as input two graphs G₁=(V₁, E₁) and G₂=(V₂, E₂), where the number of vertices in V₁ can be assumed to be less than or equal to the number of vertices in V₂. G₁ is isomorphic to an induced subgraph of G₂ if there is an injective function f which maps the vertices of G₁ to vertices of G₂ such that for all pairs of vertices x, y in V₁, edge (x, y) is in E₁ if and only if the edge (f(x), f(y)) is in E₂. The answer to the decision problem is yes if this function f exists, and no otherwise.

This is different from the subgraph isomorphism problem in that the absence of an edge in G₁ implies that the corresponding edge in G₂ must also be absent. In subgraph isomorphism, these "extra" edges in G₂ may be present.

Computational complexity

The complexity of induced subgraph isomorphism separates outerplanar graphs from their generalization series–parallel graphs: it may be solved in polynomial time for 2-connected outerplanar graphs, but is NP-complete for 2-connected series–parallel graphs.^[1][2]

Special cases

The special case of finding a long path as an induced subgraph of a hypercube has been particularly well-studied, and is called the snake-in-the-box problem.^[3] The maximum independent set problem is also an induced subgraph isomorphism problem in which one seeks to find a large independent set as an induced subgraph of a larger graph, and the maximum clique problem is an induced subgraph isomorphism problem in which one seeks to find a large clique graph as an induced subgraph of a larger graph.

Differences with the subgraph isomorphism problem

Although the induced subgraph isomorphism problem seems only slightly different from the subgraph isomorphism problem, the "induced" restriction introduces changes large enough that we can witness differences from a computational complexity point of view.

For example, the subgraph isomorphism problem is NP-complete on connected proper interval graphs and on connected bipartite permutation graphs,^[4] but the induced subgraph isomorphism problem can be solved in polynomial time on these two classes.^[5]

Moreover, the induced subtree isomorphism problem (i.e. the induced subgraph isomorphism problem where G₁ is restricted to be a tree) can be solved in polynomial time on interval graphs, while the subtree isomorphism problem is NP-complete on proper interval graphs.^[6]

References

1. Sysło, Maciej M. (1982), "The subgraph isomorphism problem for outerplanar graphs", Theoretical Computer Science, 17 (1): 91–97, doi:10.1016/0304-3975(82)90133-5, MR 0644795.
2. Johnson, David S. (1985), "The NP-completeness column: an ongoing guide", Journal of Algorithms, 6 (3): 434–451, doi:10.1016/0196-6774(85)90012-4, MR 0800733.
3. Ramanujacharyulu, C.; Menon, V. V. (1964), "A note on the snake-in-the-box problem", Publ. Inst. Statist. Univ. Paris, 13: 131–135, MR 0172736.
4. Kijima, Shuji; Otachi, Yota; Saitoh, Toshiki; Uno, Takeaki (1 November 2012). "Subgraph isomorphism in graph classes". Discrete Mathematics. 312 (21): 3164–3173. doi:10.1016/j.disc.2012.07.010.
5. Heggernes, Pinar; van 't Hof, Pim; Meister, Daniel; Villanger, Yngve (2015-01-11). "Induced Subgraph Isomorphism on proper interval and bipartite permutation graphs" (PDF). Theoretical Computer Science. 562: 252–269. doi:10.1016/j.tcs.2014.10.002. ISSN 0304-3975.
6. Heggernes, Pinar; van 't Hof, Pim; Milanič, Martin (2013). Lecroq, Thierry; Mouchard, Laurent (eds.). "Induced Subtrees in Interval Graphs" (PDF). Proceedings of the 24th International Workshop on Combinatorial Algorithms (IWOCA). Lecture Notes in Computer Science. Berlin, Heidelberg: Springer: 230–243. doi:10.1007/978-3-642-45278-9_20. ISBN 978-3-642-45278-9.