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Parameterized Leaf Power Recognition via Embedding into Graph Products

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Abstract

The k-leaf power graph G of a tree T is a graph whose vertices are the leaves of T and whose edges connect pairs of leaves at unweighted distance at most k in T. Recognition of the k-leaf power graphs for \(k \ge 7\) is still an open problem. In this paper, we provide two algorithms for this problem for sparse leaf power graphs. Our results shows that the problem of recognizing these graphs is fixed-parameter tractable when parameterized both by k and by the degeneracy of the given graph. To prove this, we first describe how to embed a leaf root of a leaf power graph into a product of the graph with a cycle graph. We bound the treewidth of the resulting product in terms of k and the degeneracy of G. The first presented algorithm uses methods based on monadic second-order logic (\({\text{MSO}}_2\)) to recognize the existence of a leaf power as a subgraph of the graph product. Using the same embedding in the graph product, the second algorithm presents a dynamic programming approach to solve the problem and provide a better dependence on the parameters.

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Acknowledgements

This research was supported in part by NSF Grants CCF-1618301 and CCF-1616248.

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  1. Department of Computer Science, University of California, Irvine, CA, USA

    David Eppstein & Elham Havvaei

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Correspondence to Elham Havvaei.

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Eppstein, D., Havvaei, E. Parameterized Leaf Power Recognition via Embedding into Graph Products. Algorithmica 82, 2337–2359 (2020). https://doi.org/10.1007/s00453-020-00720-8

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