Closed models, strongly connected components and Euler graphs

TL;DR

This paper develops a closed model in directed graphs to characterize strongly connected components and interprets Eulerian graphs within this framework, linking graph theory with homotopical concepts.

Contribution

It introduces a novel closed model for directed graphs that captures strongly connected components and relates Eulerian graphs to cofibrant objects.

Findings

Characterizes strongly connected components via a closed model.

Shows Euler graphs are cofibrant objects in this model.

Provides a cohomological proof of Euler's classical theorem.

Abstract

In this paper, we continue our study of closed models defined in categories of graphs. We construct a closed model defined in the cat-egory of directed graphs which characterizes the strongly connected components. This last notion has many applications, and it plays an important role in the web search algorithm of Brin and Page, the foun-dation of the search engine Google. We also show that for this closed model, Euler graphs are particular examples of cofibrant objects. This enables us to interpret in this setting the classical result of Euler which states that a directed graph is Euleurian if and only if the in degree and the out degree of every of its nodes are equal. We also provide a cohomological proof of this last result.