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Injective Hulls of Various Graph Classes

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Abstract

A graph is Helly if its disks satisfy the Helly property, i.e., every family of pairwise intersecting disks in G has a common intersection. It is known that for every graph G, there exists a unique smallest Helly graph \(\mathcal {H}(G)\) into which G isometrically embeds; \(\mathcal {H}(G)\) is called the injective hull of G. Motivated by this, we investigate the structural properties of the injective hulls of various graph classes. We say that a class of graphs \(\mathcal {C}\) is closed under Hellification if \(G \in \mathcal {C}\) implies \(\mathcal {H}(G) \in \mathcal {C}\). We identify several graph classes that are closed under Hellification. We show that permutation graphs are not closed under Hellification, but chordal graphs, square-chordal graphs, and distance-hereditary graphs are. Graphs that have an efficiently computable injective hull are of particular interest. A linear-time algorithm to construct the injective hull of any distance-hereditary graph is provided and we show that the injective hull of several graphs from some other well-known classes of graphs are impossible to compute in subexponential time. In particular, there are split graphs, cocomparability graphs, and bipartite graphs G such that \(\mathcal {H}(G)\) contains \(\Omega (a^{n})\) vertices, where \(n=|V(G)|\) and \(a>1\).

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Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Notes

  1. To do so, start with the set V(G) and, for each vertex v, call Refine(N[v]) for true twins or Refine(N(v)) for false twins.

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The authors declare that no funds, grants, or other support were received during the preparation of this manuscript.

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  1. Heather M. Guarnera

    Present address: The College of Wooster, Wooster, USA

Authors and Affiliations

  1. Department of Computer Science, Kent State University, Kent, USA

    Heather M. Guarnera & Feodor F. Dragan

  2. Department of Computer Science, Central Washington University, Seattle, USA

    Arne Leitert

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  1. Heather M. Guarnera
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  2. Feodor F. Dragan
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Correspondence to Heather M. Guarnera.

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Guarnera, H.M., Dragan, F.F. & Leitert, A. Injective Hulls of Various Graph Classes. Graphs and Combinatorics 38, 112 (2022). https://doi.org/10.1007/s00373-022-02512-z

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