# A Model of Directed Graph Cofiber

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## Abstract

**:**

## 1. Introduction

## 2. The Category of Directed Graphs

**Definition**

**1**

**directed graph**(or

**digraph**for short) $\overrightarrow{G}$ is a pair $(V,E)$ consisting of a set of vertices V and a set E of ordered pairs of distinct vertices in V called edges.

**Definition**

**2.**

**n-step line digraph**is a digraph consisting of vertices labeled by 0, 1, ⋯, n and exactly one edge at every two consecutive vertices, i.e., either $(i-1,i)$ or $(i,i-1)$ for all $i\in {\mathbb{Z}}^{+}$ such that $1\le i\le n$.

**Notation**

**1.**

**Definition**

**3.**

**digraph map**, $f:\overrightarrow{G}\to \overrightarrow{H}$, is a function from the vertex set of $\overrightarrow{G}$ to the vertex set of $\overrightarrow{H}$ such that whenever $(x,y)$ is an edge in $\overrightarrow{G}$ either $f\left(x\right)=f\left(y\right)$ in $\overrightarrow{H}$ or $(f\left(x\right),f\left(y\right))$ is an edge in $\overrightarrow{H}$.

**Definition**

**4.**

**image digraph**$Im\left(f\right)$ of a digraph map $f:\overrightarrow{G}\to \overrightarrow{H}$ is a directed graph consisting of a vertex set $f\left({V}_{\overrightarrow{G}}\right)$ and an edge set ${E}_{\overrightarrow{G}}=\{(f\left(x\right),f\left(y\right)):x,y\in {V}_{\overrightarrow{G}},f\left(x\right)\ne f\left(y\right),\phantom{\rule{4.pt}{0ex}}and\phantom{\rule{4.pt}{0ex}}(x,y)\in {E}_{\overrightarrow{G}}\}$.

**Definition**

**5.**

**category of directed graphs**$\mathcal{D}$ is a category in which the objects are directed graphs, $\overrightarrow{G}$, and the morphisms are digraph maps, $f:\overrightarrow{G}\to \overrightarrow{H}$.

## 3. Operations in Directed Graphs

**Definition**

**6.**

**sub-digraph**$\overrightarrow{X}$ of a digraph $\overrightarrow{G}$ denoted $\overrightarrow{X}\subset \overrightarrow{G}$ is a digraph for which ${V}_{\overrightarrow{X}}\subset {V}_{\overrightarrow{G}}$ and ${E}_{\overrightarrow{X}}\subset {E}_{\overrightarrow{G}}$.

**Definition**

**7.**

**induced sub-digraph**$\overrightarrow{X}$ of a digraph $\overrightarrow{G}$ denoted $\overrightarrow{X}\u228f\overrightarrow{G}$ is a sub-digraph in which whenever $u,v\in {V}_{\overrightarrow{X}}$ and $(u,v)\in {E}_{\overrightarrow{G}}$, then $(u,v)\in {E}_{\overrightarrow{X}}$ as well.

**Definition**

**8.**

**vertex boundary**of a sub-digraph $\overrightarrow{X}\subset \overrightarrow{G}$ is

**Definition**

**9.**

**intersection**of digraphs $\overrightarrow{G}$ and $\overrightarrow{H}$, denoted by $\overrightarrow{G}\cap \overrightarrow{H}$, is the digraph consisting of ${V}_{\overrightarrow{G}\cap \overrightarrow{H}}={V}_{\overrightarrow{G}}\cap {V}_{\overrightarrow{H}}$ and ${E}_{\overrightarrow{G}\cap \overrightarrow{H}}={E}_{\overrightarrow{G}}\cap {E}_{\overrightarrow{H}}$.

**Definition**

**10.**

**union**of digraphs $\overrightarrow{G}$ and $\overrightarrow{H}$, denoted by $\overrightarrow{G}\cup \overrightarrow{H}$, is the digraph consisting of ${V}_{\overrightarrow{G}\cup \overrightarrow{H}}={V}_{\overrightarrow{G}}\cup {V}_{\overrightarrow{H}}$ and ${E}_{\overrightarrow{G}\cup \overrightarrow{H}}={E}_{\overrightarrow{G}}\cup {E}_{\overrightarrow{H}}$.

**Definition**

**11.**

**disjoint union**of two digraphs $\overrightarrow{G}$ and $\overrightarrow{H}$, denoted $\overrightarrow{G}\coprod \overrightarrow{H}$ is given by the disjoint union of their respective vertex sets and edge sets, as sets.

**Definition**

**12.**

**graph Cartesian product**□ of two directed graphs $\overrightarrow{G}$ and $\overrightarrow{H}$ is a directed graph $\overrightarrow{G}\square \overrightarrow{H}$, where the vertices are all ordered pairs $(u,v)$ such that $u\in {V}_{\overrightarrow{G}}$ and $v\in {V}_{\overrightarrow{H}}$, and $({u}_{1},{v}_{1})\to ({u}_{2},{v}_{2})$ is an edge in $\overrightarrow{G}\square \overrightarrow{H}$ if either ${u}_{1}={u}_{2}$ and ${v}_{1}\to {v}_{2}$ in $\overrightarrow{H}$, or ${u}_{1}\to {u}_{2}$ in $\overrightarrow{G}$ and ${v}_{1}={v}_{2}$.

**Remark**

**1.**

**Definition**

**13.**

**identification digraph**resulting from the relation ∼ is a digraph $\overrightarrow{G}\coprod \overrightarrow{H}/\sim $ whose vertices are equivalence classes and whose edges are the edges between the representatives of the classes.

**Definition**

**14.**

**quotient digraph**$\overrightarrow{G}/\overrightarrow{X}$, for $\overrightarrow{X}\subset \overrightarrow{G}$ and $\overrightarrow{X}$ not necessarily connected, is an identification digraph $\overrightarrow{G}\coprod \ast /\sim $ where $x\sim \ast $ for all $x\in {V}_{\overrightarrow{X}}$.

## 4. Homotopy for Digraphs

**Definition**

**15.**

**homotopic**, denoted $f\simeq g$, if there exists an $n\ge 1$ and a digraph map $F:\overrightarrow{G}\square {I}_{n}\to \overrightarrow{H}$, for some line digraph ${I}_{n}\in {\mathcal{I}}_{n}$ (recall Notation 1), such that $F{\mid}_{\overrightarrow{G}\times \left\{0\right\}}=f$ and $F{\mid}_{\overrightarrow{G}\times \left\{n\right\}}=g$.

**Definition**

**16.**

**homotopically equivalent**(or to be of the same homotopy type) if there exist two digraph maps, $g:\overrightarrow{G}\to \overrightarrow{H}$ and $h:\overrightarrow{H}\to \overrightarrow{G}$, such that $h\circ g\simeq {id}_{\overrightarrow{G}}$ and $g\circ h\simeq {id}_{\overrightarrow{H}}$.

**Example**

**1.**

**Definition**

**17.**

**contractible**if there exists a homotopy between ${id}_{\overrightarrow{G}}$ and a constant digraph map.

**Definition**

**18.**

**homotopy category of directed graphs**, denoted $\mathrm{Ho}\mathcal{D}$, is a category in which the objects are directed graphs and the morphisms are equivalence classes of digraph maps where $f\simeq g$ whenever f and g are homotopic.

## 5. Main Construction

**Definition**

**19.**

**mapping cylinder of a digraph map**$f:\overrightarrow{G}\to \overrightarrow{H}$ is given by

**Definition**

**20.**

**extension**of a mapping cylinder ${\overrightarrow{M}}_{f}$ for a digraph map $f:\overrightarrow{G}\to \overrightarrow{H}$ is the digraph ${\overrightarrow{E}}_{f}=({V}_{{\overrightarrow{E}}_{f}},{E}_{{\overrightarrow{E}}_{f}})$ where,

**Definition**

**21.**

**extended mapping cylinder**for a digraph map $f:\overrightarrow{G}\to \overrightarrow{H}$ is given by the digraph ${\overrightarrow{EM}}_{f}={\overrightarrow{M}}_{f}\cup {\overrightarrow{E}}_{f}$.

**Definition**

**22.**

**cone**$C\overrightarrow{G}$ over a digraph $\overrightarrow{G}$ is the digraph $[\overrightarrow{G}\square {I}^{-}]/$∼, where $(g,0)\sim \ast $ for all $g\in {V}_{\overrightarrow{G}}.$

**Definition**

**23.**

**extension**of a cone over a sub-digraph $\overrightarrow{X}\subset \overrightarrow{H}$ is the digraph ${\overrightarrow{B}}_{\overrightarrow{X}}=({V}_{{\overrightarrow{B}}_{\overrightarrow{X}}},{E}_{{\overrightarrow{B}}_{\overrightarrow{X}}})$ where ${V}_{{\overrightarrow{B}}_{\overrightarrow{X}}}=\partial \overrightarrow{X}\cup \left\{p\right\}$, p is the cone point, and $(p,y)\in {E}_{{\overrightarrow{B}}_{\overrightarrow{X}}}$ if and only if $(x,y)\in {E}_{\overrightarrow{H}}$ for some $x\in \overrightarrow{X}$ and $y\in \partial \overrightarrow{X}$ or $(y,p)\in {E}_{{\overrightarrow{B}}_{\overrightarrow{X}}}$ if and only if $(y,x)\in {E}_{\overrightarrow{H}}$ for some $x\in \overrightarrow{X}$ and $y\in \partial \overrightarrow{X}$.

**Definition**

**24.**

**extended cone**over the sub-digraph $\overrightarrow{X}$ in $\overrightarrow{H}$ is $C\overrightarrow{X}\cup {\overrightarrow{B}}_{\overrightarrow{X}}$.

**Construction**

**1.**

**digraph cofiber**$\overrightarrow{C}\left(f\right)$ for a map $f:\overrightarrow{G}\to \overrightarrow{H}$ is an extension of a cone over $\overrightarrow{G}\coprod Im\left(f\right)$ (viewed as an induced subdigraph of ${\overrightarrow{M}}_{f}$) unioned with the identification digraph $\overrightarrow{G}\square {I}^{+}\coprod \overrightarrow{H}/\sim $ where $(g,0)\sim f\left(g\right)$ and the extension of this reversed mapping cylinder (See Figure 1).

**Remark**

**2.**

**Theorem**

**1.**

**Proof.**

## 6. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Gurevich, B.M. On classes of infinite loaded graphs with randomly deleted edges. Appl. Math. Nonlinear Sci.
**2020**, 5, 257–260. [Google Scholar] [CrossRef] - Hosamani, S.M.; Awati, V.B.; Honmore, R.M. On graphs with equal dominating and c-dominating energy. Appl. Math. Nonlinear Sci.
**2019**, 4, 503–512. [Google Scholar] [CrossRef] [Green Version] - Grigor’yan, A.; Lin, Y.; Muranov, Y.; Yau, S.-T. Homologies of path complexes and digraphs. arXiv
**2012**, arXiv:1207.2834. [Google Scholar] - Grigor’yan, A.; Lin, Y.; Muranov, Y.; Yau, S.-T. Cohomology of digraphs and (undirected) graphs. Asian J. Math.
**2015**, 19, 887–931. [Google Scholar] [CrossRef] [Green Version] - Grigor’yan, A.; Muranov, Y.; Yau, S.-T. On a cohomology of digraphs and Hochschild cohomology. J. Homotopy Relat. Struct.
**2016**, 11, 209–230. [Google Scholar] [CrossRef] - Grigor’yan, A.; Jimenez, R.; Muranov, Y.; Yau, S.-T. On the path homology theory of digraphs and Eilenberg-Steenrod axioms. Homol. Homotopy Appl.
**2018**, 20, 179–205. [Google Scholar] [CrossRef] - Grigor’yan, A.; Muranov, Y.; Yau, S.-T. Homologies of digraphs and Künneth formulas. Comm. Anal. Geom.
**2017**, 25, 969–1018. [Google Scholar] [CrossRef] - Grigor’yan, A.; Muranov, Y.; Vershinin, V.; Yau, S.-T. Path homology theory of multigraphs and quivers. Forum Math.
**2018**, 30, 1319–1337. [Google Scholar] [CrossRef] - Lin, Y.; Wang, C.; Yau, S.-T. Discrete Morse Theory on Digraphs. arXiv
**2021**, arXiv:2102.10518. [Google Scholar] - Gianella, G.M. Su una omotopia regolare dei grafi. Rend. Sem. Mat. Univ. Politec. Torino
**1976**, 35, 349–360. [Google Scholar] - Malle, G. A homotopy theory for graphs. Glas. Mat. Ser. III
**1983**, 18, 3–25. [Google Scholar] - Chen, B.; Yau, S.-T.; Yeh, Y.-N. Graph homotopy and Graham homotopy. Discret. Math.
**2001**, 241, 153–170. [Google Scholar] [CrossRef] [Green Version] - Grigor’yan, A.; Lin, Y.; Muranov, Y.; Yau, S.-T. Homotopy theory for digraphs. Pure Appl. Math. Q.
**2014**, 10, 619–674. [Google Scholar] [CrossRef] [Green Version] - Brown, E.H., Jr. Cohomology theories. Ann. Math.
**1962**, 75, 467–484. [Google Scholar] [CrossRef] - Brown, E.H., Jr. Abstract homotopy theory. Trans. Amer. Math. Soc.
**1965**, 119, 79–85. [Google Scholar] [CrossRef] - McGuirk, Z.; Park, B. Brown representability for directed graphs. arXiv
**2021**, arXiv:2003.07426. [Google Scholar] - Sampietro, J. Homotopy Theory of Digraphs—A Categorical Viewpoint. MCA Special Session: Categories and Topology. 2021. Available online: https://www.youtube.com/watch?v=C_O92uCk1Qo (accessed on 20 December 2021).
- Adams, J.F. A variant of EH Brown’s representability theorem. Topology
**1971**, 10, 185–198. [Google Scholar] [CrossRef] [Green Version]

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McGuirk, Z.; Park, B.
A Model of Directed Graph Cofiber. *Axioms* **2022**, *11*, 32.
https://doi.org/10.3390/axioms11010032

**AMA Style**

McGuirk Z, Park B.
A Model of Directed Graph Cofiber. *Axioms*. 2022; 11(1):32.
https://doi.org/10.3390/axioms11010032

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McGuirk, Zachary, and Byungdo Park.
2022. "A Model of Directed Graph Cofiber" *Axioms* 11, no. 1: 32.
https://doi.org/10.3390/axioms11010032