Combinatorial properties of continuous graphs: A survey of challenges, solutions and open problems

TL;DR

This survey explores the complexities and open challenges of combinatorial optimization problems on continuous or metric graphs, where edges are represented by intervals, highlighting real-world relevance and unresolved questions.

Contribution

It provides a comprehensive review of existing results and identifies open problems in optimization on continuous graphs, a less-studied extension of classical graph problems.

Findings

Many classical problems become more complex on continuous graphs.

Open problems remain in the optimization of continuous graph properties.

Real-world applications often favor continuous graph models.

Abstract

Inspired by notorious combinatorial optimization problems on graphs, in this paper we consider a series of related problems defined using a metric space and topology determined by a graph. Particularly, we present the Independent Set, Vertex Cover, Chromatic Number and Treewidth problems on, so-called, continuous or metric graphs where every edge is represented by a unit-length continuous interval rather than by a pair of vertices. If any point of any unit-interval edge is considered as a possible member of a hitting set or a cover, the classical combinatorial problems become trickier and many open questions arise. Notably, in many real-life applications, such a continuous view of a graph is more natural than the classic combinatorial definition of a graph. The contribution of this paper is twofold: i) we survey the known results for optimization problems on continuous graphs, and ii) we create a list of open problems related to the continuous graphs.