Injective Hulls of Various Graph Classes

• Published in: Graphs and combinatorics Vol. 38; no. 4
• Main Authors: Guarnera, Heather M., Dragan, Feodor F., Leitert, Arne
• Format: Journal Article
• Language: English
• Published: Tokyo Springer Japan 01.08.2022; Springer Nature B.V
• Subjects: Algorithms; Apexes; Combinatorics; Disks; Engineering Design; Graphs; Hulls; Mathematics; Mathematics and Statistics; Original Paper; Permutations; Helly graphs; Injective hull; hyperbolic; Square-chordal graphs; Permutation graphs; Distance-hereditary graphs; Chordal graphs
• ISSN: 0911-0119; 1435-5914
• DOI: 10.1007/s00373-022-02512-z

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, and bipartite graphs G such that H ( G ) contains Ω ( a n ) vertices, where n = | V ( G ) | and a > 1 .