Injective hulls of various graph classes

TL;DR

This paper studies the structural properties of injective hulls of various graph classes, identifying which classes are closed under this operation and providing algorithms for their computation.

Contribution

It characterizes classes closed under Hellification, proves that some classes are not, and offers a linear-time algorithm for the injective hull of distance-hereditary graphs.

Findings

Chordal, square-chordal, and distance-hereditary graphs are closed under Hellification.

Permutation graphs are not closed under Hellification.

Efficient algorithms exist for certain classes, but computing injective hulls can be infeasible for others.

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 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 $\mathcal{C}$ is closed under Hellification if $G \in \mathcal{C}$ implies $H(G) \in \mathcal{C}$. 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, bipartite graphs G such that H(G) contains $\Omega(a^{n})$ vertices, where $n=|V(G)|$ and $a>1$.