Injective hulls of various graph classes

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 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, squarechordal 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.

Beyond Helly graphs: the diameter problem on absolute retracts

“…We stress that the proof of Theorem 10 is constructive, and that it leads to a polynomialtime algorithm in order to construct an absolute planar retract in which the input planar graph G isometrically embeds. In contrast to our result, the smallest Helly graph in which a graph G isometrically embeds may be exponential in its size [59].…”