Parameterized Complexity of Induced Graph Matching on Claw-Free Graphs

Abstract

The Induced Graph Matching problem asks to find \(k\) disjoint induced subgraphs isomorphic to a given graph \(H\) in a given graph \(G\) such that there are no edges between vertices of different subgraphs. This problem generalizes the classical Independent Set and Induced Matching problems, among several other problems. We show that Induced Graph Matching is fixed-parameter tractable in \(k\) on claw-free graphs when \(H\) is a fixed connected graph, and even admits a polynomial kernel when \(H\) is a complete graph. Both results rely on a new, strong, and generic algorithmic structure theorem for claw-free graphs. Complementing the above positive results, we prove \(\mathsf {W}[1]\)-hardness of Induced Graph Matching on graphs excluding \(K_{1,4}\) as an induced subgraph, for any fixed complete graph \(H\). In particular, we show that Independent Set is \(\mathsf {W}[1]\)-hard on \(K_{1,4}\)-free graphs. Finally, we consider the complexity of Induced Graph Matching on a large subclass of claw-free graphs, namely on proper circular-arc graphs. We show that the problem is either polynomial-time solvable or \(\mathsf {NP}\)-complete, depending on the connectivity of \(H\) and the structure of \(G\).

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Notes

  1. 1.

    Observe that the reduction given here does not work in the case that \(H\) is a triangle, because the pre-image of a triangle is not unique. The triangle is also the only line graph for which the pre-image is not unique [43], which is why the reduction works for all other cases. We give a proof of the \(\mathsf {NP}\)-hardness of Induced Graph Matching for \(H = K_3\) in the Appendix.

  2. 2.

    We can explicitly prove this using the language of Sect. 4. Note that any co-bipartite graph is a thickening of a single semi-edge, a single vertex, or two independent vertices. Moreover, the graph consisting of a single semi-edge, a single vertex, or two independent vertices is a circular interval trigraph. Hence, using Lemma 1, we can see that any co-bipartite graph is a fuzzy circular-arc graph.

  3. 3.

    We note that the definition of isomorphism can be easily extended to hypergraphs.

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Acknowledgments

We thank the anonymous reviewers for helpful remarks improving the presentation of this manuscript.

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Correspondence to Erik Jan van Leeuwen.

Additional information

An extended abstract of the results in this paper have appeared in the Proceedings of 20th European Symposium on Algorithms (ESA 2012) [26]. Partially supported by ERC StG project PAAl No. 259515. The research leading to these results has received funding from the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007–2013) under REA grant agreement No. 631163.11.

Appendix: Induced Graph Matching for Triangles is \(\mathsf {NP}\)-hard on Line Graphs

Appendix: Induced Graph Matching for Triangles is \(\mathsf {NP}\)-hard on Line Graphs

Recall from the introduction that the immediate correspondence between an \(H\)-matching in a graph \(G\) and an induced \(L(H)\)-matching in \(L(G)\) does not apply in case \(L(H)\) is a triangle, i.e. in case that \(H\) is \(K_{3}\) or \(K_{1,3}\). Hence, to the best of our knowledge, the complexity of Induced Graph Matching for \(H = K_3\) (also known as Induced Triangle Matching or Induced Triangle Packing) on line graphs is open.

In this section, we prove that Induced Graph Matching for \(H = K_3\) on line graphs is \(\mathsf {NP}\)-hard.

Theorem 15

Induced Graph Matching for \(H = K_3\) is \(\mathsf {NP}\)-complete on line graphs of planar graphs.

Proof

It is clear that Induced Graph Matching for \(H = K_3\) on line graphs of planar graphs belongs to \(\mathsf {NP}\). To prove \(\mathsf {NP}\)-hardness, we reduce from Independent Set on planar graphs where each vertex has degree exactly three, which is known to be \(\mathsf {NP}\)-complete [34]. Let \((G,k)\) be an instance of this problem. We transform this to an instance of Induced Graph Matching with \(H = K_{3}\) as follows. First, we subdivide each edge of \(G\). That is, we (simultaneously) remove each edge \(e = \{u,v\}\) and add a new vertex \(x_{e}\) and new edges \(\{u,x_{e}\}\) and \(\{x_{e},v\}\). Denote the resulting graph by \(G'\). Then the instance of Induced Graph Matching is \((L(G'), K_{3}, k)\), where \(L(G')\) is the line graph of \(G'\). Note that \(G'\) is planar, and thus \(L(G')\) is the line graph of a planar graph.

Observe that any induced subgraph in \(L(G')\) that is isomorphic to \(K_{3}\) corresponds to three edges of \(G'\) that are incident on the same vertex. Armed with this observation, we show that \((L(G'), K_{3}, k')\) is a “yes”-instance of Induced Graph Matching if and only if \((G,k)\) is a “yes”-instance of Independent Set, thus completing the proof.

Suppose that \((G,k)\) is a “yes”-instance of Independent Set. Let \(I\) be an independent set of \(G\) of size \(k\). For each \(v \in I\), consider the three edges of \(G'\) that are incident on \(v\). These correspond to a triangle in \(L(G')\). Let \(M\) be the set of these triangles for all vertices in \(I\). Note that \(|M| = k\). By the above observation, no two triangles in \(L(G')\) have a vertex in common. Moreover, since \(I\) is independent, no two triangles in \(M\) are adjacent. Therefore, \((L(G'), K_{3}, k)\) is a “yes”-instance of Induced Graph Matching.

Suppose that \((L(G'), K_{3}, k)\) is a “yes”-instance of Induced Graph Matching. Let \(M\) be an induced \(K_{3}\)-matching in \(L(G')\) of size \(k\). By the above observation, no two triangles in \(L(G')\) have a vertex in common, and by construction each triangle corresponds to some vertex \(v\) that is both in \(G\) and \(G'\). Let \(I\) be the set of vertices that correspond to the triangles of \(M\). Since \(M\) cannot contain triangles that correspond to adjacent vertices in \(G\), \(I\) is an independent set. As \(|I|=|M|=k\), \((G,k)\) is a “yes”-instance of Independent Set. \(\square \)

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Hermelin, D., Mnich, M. & van Leeuwen, E.J. Parameterized Complexity of Induced Graph Matching on Claw-Free Graphs. Algorithmica 70, 513–560 (2014). https://doi.org/10.1007/s00453-014-9877-5

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Keywords

  • Algorithms and data structures
  • Induced matchings
  • Claw-free graphs
  • Fixed-parameter tractability