In this article, we introduce a categorical characterization of directed and undirected graphs, and explore subcategories of reflexive and simple graphs. We show that there are a number of adjunctions between such subcategories, exploring varying combinations of graph types

Ender, Jacob D

English

Scholarship@Western

Categorical Aspects of GraphsJacob Ender - Western USRIAt their most basic, graphs consist of a set of vertices and a set of edges. Graphscan be directed (i.e. the edges point from one node to another), or undirected,with no prescribed flow. This section concerns categorical aspects of graphs.We introduce the category of directed graphs and then explore various subcat-egories, considering graphswith different combinations of properties. Namely,we discuss graphs which are directed, undirected, reflexive, simple, and com-binations thereof.1. The Category of Directed GraphsDefinition 1.1 (G). Define a category G to be the category+ BCwith two objects + and  and two non-identity morphisms.Observe that a presheaf  on this category consists of two sets, whichwewillcall + and , together with two functions B∗ , C∗ :  → + , noting thatcontravariance necessitates reversing the arrows. This structure is precisely adirected graph, where + is the set of vertices,  the set of edges and B∗ and C∗are the source and target functions, respectively. We may thus view a directedgraph as a presheaf  : Gop → Set, or equivalently an object in the presheafcategory SetGop.Definition 1.2 (Category of Directed Graphs; DirGrph). Define the category ofdirected graphs DirGrph to be the category SetGop. We have already establishedobjects in this category, but it remains to determine the morphisms. Evidently,morphisms of graphs are natural transformations, since the graphs themselvesare functors. This means that a morphism 5 : →  of directed graphs = ( , + , B∗ , C∗)and = ( , + , B ∗ , C ∗)consists of functions 5+ : + → + and 5 :  →  such that5+ ◦ B∗ = B ∗ ◦ 5and5+ ◦ C∗ = C ∗ ◦ 5as illustrated by the commutative square below:Jacob Ender + +C∗B∗5C∗B∗5+2. The Category of Simple Directed GraphsHaving developed the category of directed graphs, we now turn our atten-tion to some of its subcategories, as well as some nice properties that thesesubcategories have. We begin with the category of simple directed graphs.Definition 2.1 (Simple Directed Graph). A simple directed graph is a directedgraph in which there is at most one edge between any pair of vertices.Given some graph  ∈ DirGrph, there is an induced morphism〈B∗ , C∗〉 :  → + × +taking an edge 4 ∈  to the pair (B∗(4), C∗(4)). It is easy to see that is simpleprecisely when this map is a monomorphism, as each edge has a unique valueunder 〈B∗ , C∗〉 in a simple directed graph.As simple directed graphs form a subcategory ofDirGrph, there is an evidentinclusion functor * : SimpDirGrph ↩→ DirGrph. It is also possible to create asimple directed graph from any directed graph by “collapsing” each edge toone single edge. In fact, we claim that the functor realizing this process definesa left adjoint to the inclusion functor.Proposition 2.1. The inclusion functor * : SimpDirGrph ↩→ DirGrph admits aleft adjoint  : DirGrph → SimpDirGrph, and thus SimpDirGrph is a reflectivesubcategory of DirGrph.The proof relies on the following lemma.Lemma 2.1.(i) Given two graphs andwith simple, amap 5 : →  is completelydetermined by 5+ : + → + .(ii) A map ! : + → + is part of a graph map 5 = (!, !′) if and only iffor all E, E′ ∈ + , we have E ∼ E′ ⇒ !(E) ∼ !(E′). That is, adjacentvertices are mapped to adjacent vertices.Proof. The first part of the lemma is clear, as when a graph is simple one hasno choice of which edges to map to while respecting adjacency, since simplegraphs may have at most one edge between pairs of vertices.Suppose now that ! : + → + is part of a graph map 5 = (!, !′), andsuppose that E ∼ E′ for vertices E, E′ ∈ + . Then there is an edge 4 ∈  such2Jacob Enderthat B∗(4) = E and C∗(4) = E′. Observe that by commutativity of ! and !′with the source and target functions, we have thatB∗(!′(4)) = !(B∗(4)) = !(E)andC∗(!′(4)) = !(C∗(4)) = !(E′).Thus there is an edge between !(E) and !(E′), so !(E) ∼ !(E′). Conversely,suppose that for all E, E′ ∈ + we have E ∼ E′⇒ !(E) ∼ !(E′). Take an edge4 ∈  and define E := B∗(4) and E′ := C∗(4). Thus E ∼ E′, so by assumptionwe have !(E) ∼ !(E′). We then have an edge 4 ∈  between !(E) and !(E′)with B ∗(4) = !(E) and C ∗(4) = !(E′). Now, define !′(4) = 4. To check therequired commutativity, observe thatB∗(!′(4)) = !(E) = !(B∗(4))as required, and a similar calculation is made to check commutativity with thetarget. Thus 5 := (!, !′) forms a graph map. Proof of Proposition 2.1. We begin by defining the action of  on the vertexand edge sets of graphs. For  ∈ DirGrph, define ()+ = + , and define() = Im(〈B∗ , C∗〉). We claim that  is left adjoint to * ; that is, we have theisomorphismDirGrph(,*)  SimpDirGrph(, ).Consider some graph map 5 :  → *. As * =  by definition, we havea map 5 :  → . By the first part of Lemma 2.1, 5 is completely determinedby 5+ : + → + . Because ()+ = + , we have 5+ : ()+ → + . ByLemma 2.3(ii), 5+ extends to a graph map if E ∼ E′ ⇒ 5+ (E) ∼ 5+ (E′) for allE, E′ ∈ . Since E ∼ E′ if and only if E ∼ E′, the result follows by 5 beinga graph map. Thus we obtain a graph map 6 :  → , giving the desiredisomorphism. We have now established a reflective subcategory of DirGrph, but similaradjunctions exist between subcategories with different graph properties. Next,we turn our focus to reflexive directed graphs.3. The Category of Reflexive Directed GraphsWe have not yet explored the case in which graphs have loops on each vertex. Itturns out that graphs with this property form another subcategory of DirGrph,which also admits some nice properties that we investigate further in thissection. Graphs with loops on vertices are called reflexive graphs.Definition 3.1 (Reflexive Directed Graph). A reflexive directed graph is a di-rected graph in which each vertex has a distinguished loop on it. Equivalently,for each vertex in the graph, there exists a distinguished edge from that vertexto itself.3Jacob EnderWe build the category of reflexive directed graphs as a presheaf category,but with some additional structure.Definition 3.2 (G+). Define a category G+ to be the category+ BCAwith two objects and three non-identity morphisms subject to the conditionAB = id+ = AC.In the same spirit as Definition 1.2, we define the category of reflexive di-rected graphs, RelfDirGrph, to be the category SetGop+ . This category is quitesimilar to the category of directed graphs, with the addedmorphism A witness-ing the loops on each vertex.Note that there is the evident inclusion functor 8 : G ↩→ G+ which inducesa functor 8∗ : SetGop+ → SetGop by precomposition:Gop Gop+ Set8op That is, 8∗ =  ◦ 8op. We claim that this functor has a right adjoint, which takesa directed graph and freely adds loops to each vertex.Proposition 3.1. The functor 8∗ : SetGop+ → SetGop has a right adjoint 8∗ :SetGop → SetGop+ .Proof. Define (8∗)+ = + and (8∗) = + t  for a graph  ∈ DirGrph. Wedefine the action of 8∗ on morphisms as follows. For the source mapB(8∗)∗ : + t  → +define for all 4 ∈ + t B(8∗)∗ ={B∗ if 4 ∈ 4 if 4 ∈ +and similarly for the target map. The map A(8∗)∗ : + → + t  is simplydefined by the coproduct inclusion. We wish to establish the isomorphismReflDirGrph(, 8∗)  DirGrph(8∗, ).Observe that amap 5 : → 8∗ of reflexive graphs consists of maps 5+ : + →+ and 5 : + → + t . Conversely, a map 6 : 8∗→  consists of maps6+ : + → +and6 :  →  .4Jacob EnderIt is clear that the vertex maps are in one-to-one correspondence. Notice,however, that the graph  lives in ReflDirGrph, so  is in fact a reflexive graph.Thus amap 6 :  →  is precisely amap 5 :  → +t after forgettingabout the loops on each vertex. This gives the correspondence between edgemaps, completing the isomorphism. Wecontinue our study of reflexive directed graphs in the next section, wherewe investigate simple reflexive directed graphs.4. The Category of Simple Reflexive Directed GraphsThe final subcategory of DirGrph to investigate is the category of simple re-flexive directed graphs, which we denote SimpReflDirGrph. First, observe thatSimpReflDirGrph ⊆ SimpDirGrph, and we have the same pre-composition func-tor 8∗ : SimpReflDirGrph → SimpDirGrph as defined in Section 3. It is naturalto ask again what kinds of adjunctions this functor admits when we restrictourselves to simple graphs. If we wish to “reflexify” a graph while keepingit simple, we ultimately have two choices. We may simply remove all verticeswhich do not have loops, or we may add a loop wherever there was not one,and preserve all other edges. It turns out that these two processes assembleneatly into left and right adjoints to the functor 8∗. We first examine the rightadjoint, which removes those vertices which do not have loops.Proposition 4.1. The functor 8∗ : SimpReflDirGrph → SimpDirGrph admits aright adjoint 8∗ : SimpDirGrph→ SimpReflDirGrph.Proof. Define (8∗)+ = {E ∈ + | (E, E) ≠ ∅}. That is, we define (8∗)+ tobe the set of vertices in + which have a loop. Now, define (8∗)(E, E′) =(E, E′) for all E, E′ ∈ + . The unit of the proposed adjunction is a naturaltransformation  : idSimpReflDirGrph ⇒ 8∗8∗. Components are morphisms  : → 8∗8∗, and since 8∗ = , components are morphisms  :  → 8∗. But,since G is reflexive already, 8∗ = , andwemay thus define  = id. Turningto the counit & : 8∗8∗ ⇒ idSimpDirGrph, observe that components are morphisms& : 8∗(8∗) → , or equivalently morphisms & : 8∗→ . Thus, we define &to be the embedding of 8∗ into . It is straightforward to check that the unitand counit satisfy the required triangle identities. This completes one half of the promise we made at the beginning of thesection. We now turn our focus to the left adjoint to 8∗, which adds loops ratherthan removing vertices which do not have them.5Jacob EnderProposition 4.2. The functor 8∗ : SimpReflDirGrph→ SimpDirGrph admits a leftadjoint 8! : SimpDirGrph→ SimpReflDirGrph.Proof. We define (8!)+ = + and(8!)(E, E′) ={{∗} if E = E′ or (E, E′) ≠ ∅∅ if (E, E′) = ∅ and E ≠ E′.That is, we preserve existing edges and add loops where there were not loopsbefore. The unit of the intended adjunction is a natural transformation  :idSimpDirGrph ⇒ 8∗8! with components  : → 8∗8!, but since 8∗(8!) = 8!, wehave components  :  → 8!. We define  to be the embedding of  into8!. The counit is a natural transformation & : 8!8∗ ⇒ idSimpReflDirGrph, and hascomponents which are morphisms & : 8!8∗() → . Similarly, since 8∗ = ,we have morphisms & : 8! → . Since  was already a simple reflexivegraph, 8! = . Thus we may define & : →  to be the identity on . As inProposition 4.1, the triangle identities are easily checked. Having built our two adjunctions, we have completed our survey of directedgraphs. The last uncharted territory is an investigation of undirected graphs,and how our constructions on directed graphs may translate to graphs withoutdirection. The final section will explore these constructions.5. The Category of Undirected Graphs and PropertiesThe first question to ask is how to formalize the notion of an undirected graphfrom a categorical standpoint. Consider the category+ BCwhich is the same category we used to build DirGrph, but equipped with an in-volution  : →  such that B = C, C = B and  ◦  = id. In this way, we canview  as a “swapper”whichmakes direction ambiguous - applying  conflatessources and targets, and witnesses our graph becoming “non-directional”. Un-surpringly, the category of undirected graphs, Grph, is the presheaf categoryon this category. Note that we can still use the source and target functionsas we did before. For example, the notion of being simple is the same: anundirected graph  is simple when the map 〈B∗ , C∗〉 :  → + × + is amonomorphism. Observe that there is an inclusion of G into this category.6Jacob EnderAnother question to ask is how to build reflexive undirected graphs. Con-sider the category+ BCAsubject to the requirements AB = id+ and AC = id+ . This is precisely thecategory G+ equipped with the involution  making our graph undirected. Asexpected, reflexive undirected graphs are presheaves on this category. It isworth observing that this category is actually a full subcategory of Set: thereis a fully faithful functor ( sending + to the set {1},  to the set {1, 2}, B to thefunction 1 : {1} → {1, 2} which picks out 1, and sends C to a similar function2 : {1} → {1, 2}. The morphism A is sent to the unique map {1, 2} → {1}, and is sent to the function on {1, 2}which swaps the argument. There is preciselythe same adjunction between SimpReflGrph and SimpGrph as therewas betweenSimpReflDirGrph and SimpDirGrph, as is easily verified by using the same adjointfunctors we defined in Section 4.7

Categorical Aspects of Graphs

https://ir.lib.uwo.ca/cgi/viewcontent.cgi?article=1181&context=usri

Categorical Aspects of Graphs

Abstract

Similar works

Full text

Available Versions

Scholarship@Western