Injective hulls of various graph classes
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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 H(G) into which G isometrically embeds; 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 𝒞 is closed under Hellification if G ∈𝒞 implies H(G) ∈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, bipartite graphs G such that H(G) contains Ω(a^n) vertices, where n=|V(G)| and a>1.
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