Compression with wildcards: All induced metric subgraphs

TL;DR

This paper introduces two algorithms for enumerating metric subgraphs in graphs, one producing compressed output and the other generating sets individually with polynomial delay, advancing graph enumeration methods.

Contribution

It presents novel algorithms for enumerating metric and geodesically-convex subgraphs, with one algorithm providing compressed enumeration and the other ensuring output-polynomial time.

Findings

The { t AllMetricSets} algorithm efficiently compresses metric subgraph enumeration.

The second algorithm guarantees output-polynomial time for listing metric sets.

The methods extend to geodesically-convex sets, strengthening the enumeration framework.

Abstract

Driven by applications in the natural, social and computer sciences several algorithms have been proposed to enumerate all sets $X\s V$ of vertices of a graph $G=(V,E)$ that induce a {\it connected} subgraph. We offer two algorithms for enumerating all $X$'s that induce (more exquisite) {\it metric} subgraphs. Specifically, the first algorithm, called {\tt AllMetricSets}, generates these $X$'s in a compressed format. The second algorithm generates all (accessible) metric sets one-by-one but is provably output-polynomial. Mutatis mutandis the same holds for the geodesically-convex sets $X\s V$, this being a natural strengthening of "metric". The Mathematica command {\tt BooleanConvert} features prominently.