Induced subgraph isomorphism problem

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Maximum lengths of snakes (Ls) and coils (Lc) in the snakes-in-the-box problem for dimensions n from 1 to 4 Snakes and coils in the box.svg
Maximum lengths of snakes (Ls) and coils (Lc) in the snakes-in-the-box problem for dimensions n from 1 to 4

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.

Contents

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]

Related Research Articles

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<span class="mw-page-title-main">Independent set (graph theory)</span> Unrelated vertices in graphs

In graph theory, an independent set, stable set, coclique or anticlique is a set of vertices in a graph, no two of which are adjacent. That is, it is a set of vertices such that for every two vertices in , there is no edge connecting the two. Equivalently, each edge in the graph has at most one endpoint in . A set is independent if and only if it is a clique in the graph's complement. The size of an independent set is the number of vertices it contains. Independent sets have also been called "internally stable sets", of which "stable set" is a shortening.

<span class="mw-page-title-main">Complete bipartite graph</span> Bipartite graph where each node of 1st set is linked to all nodes of 2nd set

In the mathematical field of graph theory, a complete bipartite graph or biclique is a special kind of bipartite graph where every vertex of the first set is connected to every vertex of the second set.

<span class="mw-page-title-main">Perfect graph</span> Graph with tight clique-coloring relation

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<span class="mw-page-title-main">Chordal graph</span> Graph where all long cycles have a chord

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<span class="mw-page-title-main">NP-completeness</span> Complexity class

In computational complexity theory, a problem is NP-complete when:

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  3. The correctness of each solution can be verified quickly and a brute-force search algorithm can find a solution by trying all possible solutions.
  4. The problem can be used to simulate every other problem for which we can verify quickly that a solution is correct. In this sense, NP-complete problems are the hardest of the problems to which solutions can be verified quickly. If we could find solutions of some NP-complete problem quickly, we could quickly find the solutions of every other problem to which a given solution can be easily verified.

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<span class="mw-page-title-main">Bipartite half</span> Graph whose nodes are one of the vertex sets of a bipartite graph

In graph theory, the bipartite half or half-square of a bipartite graph G = (U,V,E) is a graph whose vertex set is one of the two sides of the bipartition (without loss of generality, U) and in which there is an edge uiuj for each pair of vertices ui, uj in U that are at distance two from each other in G. That is, in a more compact notation, the bipartite half is G2[U] where the superscript 2 denotes the square of a graph and the square brackets denote an induced subgraph.

<span class="mw-page-title-main">Odd cycle transversal</span>

In graph theory, an odd cycle transversal of an undirected graph is a set of vertices of the graph that has a nonempty intersection with every odd cycle in the graph. Removing the vertices of an odd cycle transversal from a graph leaves a bipartite graph as the remaining induced subgraph.

In graph theory, an induced matching or strong matching is a subset of the edges of an undirected graph that do not share any vertices and these are the only edges connecting any two vertices which are endpoints of the matching edges.

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). "Induced Subtrees in Interval Graphs" (PDF). In Lecroq, Thierry; Mouchard, Laurent (eds.). Proceedings of the 24th International Workshop on Combinatorial Algorithms (IWOCA). Lecture Notes in Computer Science. Berlin, Heidelberg: Springer. pp. 230–243. doi:10.1007/978-3-642-45278-9_20. ISBN   978-3-642-45278-9.