Injective Hulls of Various Graph Classes

Document Type

Article

Department or Administrative Unit

Computer Science

Publication Date

6-29-2022

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, 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(an) vertices, where n=|V(G)| and a>1.

Comments

This article was originally published in Graphs and Combinatorics. The full-text article from the publisher can be found here.

Due to copyright restrictions, this article is not available for free download from ScholarWorks @ CWU.

Journal

Graphs and Combinatorics

Rights

Copyright © 2022, The Author(s), under exclusive licence to Springer Japan KK, part of Springer Nature

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