The Fundamental Morphism Theorem in the Categories of Graphs & Graph Reconstruction
The Fundamental Morphism Theorem is a categorical version of the First Noether Isomorphism Theorem for categories that do not have kernels or cokernels. We consider two categories of graphs. Both categories will admit graphs with multiple edges and loops, and are distinguished by allowing two different types of homomorphisms, standard graph homomorphisms and more general graph homomorphisms where the contraction of an edge is allowed. We establish the Fundamental Morphism Theorem in these two categories of graphs. We then use the result to provide an equivalent reformulation of the vertex and edge reconstruction conjectures. This reformulation shows that reconstructability is equivalent to the existence of a graph homomorphism satisfying an equation.
Keywords: graph homomorphism, graph isomorphism, category of graphs, quotient graph, reconstruction conjecture
We will follow the notations of [Bondy] for graph theory, and in specific we use as the incidence function. The exception to this is that we will name graph homomorphisms as strict graph morphisms. We use this terminology to separate strict graph morphisms from a more general graph homomorphism, graph morphisms, where edges can be mapped to vertices provided incidence is still preserved. This is not the standard graph homomorphism [HN2004], but it has two natural advantages. First, it allows the contraction of an edge to be considered as a morphism, and second, it generalizes the morphisms often studied by category theorists when considering the category of directed graphs [Goldblatt, Mac].
The aim of this paper is to establish the Fundamental Morphism Theorem as in [Lawvere] for the category of graphs with graph morphisms and the category of graphs with strict graph morphisms. In both categories we allow for graphs with multiple edges and loops. This result distinguishes these two categories, as the Fundamental Morphism Theorem often fails to hold in non-abelian categories. For example, the Fundamental Morphism Theorem fails to hold in the category of topological spaces and continuous functions. We also show that if you restrict the graphs to be simple in the graph morphism case, or simple with at most one loop allowed on a vertex in the strict graph morphism case, as in the standard category of graphs [HN2004], the Fundamental Morphism Theorem fails to hold.
Once the Fundamental Morphism Theorem is established, in section LABEL:ERC we apply it to provide a reformulation of the vertex and edge reconstruction conjectures [Harary]. In these reformulations, we only conjecture the existence of an epimorphism that satisfies a single graph homomorphism equation.
1.1 Categorical Constructions
To aid in a formal definition of a graph morphism, we define the part set of a graph to be . Given two graphs and , a graph morphism is a function with that preserves incidence, i.e. whenever , for all and some . As is a restriction of , for we will often write instead of . In this definition of morphism, edges can be mapped to vertices as long as incidence is preserved. If we add the restriction that edges must be mapped to edges, we call the resulting morphism a strict graph morphism or strict morphism.
We assume the reader is familiar with epimorphisms and momomorphisms. However, for the proof of the result we include the definition for a special type of epimorphism. A morphism is an extremal epimorphism if does not factor through any proper monomorphism, i.e. if with a monomorphism and an epimorphism, then is an isomorphism [JoyofCats].
We are concerned with four categories of graphs. We call the category of all graphs with all graph morphisms Grphs, the category of all graphs with strict graph morphisms StGrphs, and the category of simple graphs with all graph morphisms SiGrphs. When we restrict the allowed graphs for the category using strict graph morphisms, we will use simple graphs where at most a single loop is allowed on each vertex. This category will be denoted SLStGrphs, and is the standard category of graphs [HN2004]. K.K. Williams developed versions of the three Noether Isomorphism Theorems for Grphs via a concretely defined quotient graph [KKWil].
We now turn to the required categorical constructions in the four categories of graphs. Proofs that these constructions satisfy the categorical universal mapping properties are straight-forward, and more details can be found in [Plessas].
Given two graphs and in Grphs, the categorical product is an generalization of the strong product of graphs where we define by and for with and with there is an element with and if and , there is another element with that has the same projections as . In SiGrphs the categorical product is exactly the strong product. In StGrphs and SLStGrphs the categorical product is the tensor product of graphs, but for our purposes we can follow the construction of Grphs but delete all pairs if exactly one of or is a vertex.
In all four categories of graphs the coproduct, , of two graphs and is the disjoint union of the two graphs, and the equalizer, , of two morphism is the inclusion morphism of the subgraph of defined by and if then and . The incidence condition ensures that an edge is included in the equalizer only if the incident vertices are as well.
In Grphs and StGrphs the coequalizer, , of two morphism is the natural quotient morphism from to defined by where is the equivalence relation defined by if there is a sequence such that and or , where if an edge is identified with a vertex, the result is a vertex in .
In SLStGrphs we follow the same construction for the coequalizer but we also identify any parallel edges to a single edge and any multiple loops to a single loop, and in SiGrphs we also identify any loops to their incident vertex.
In a category with products, coproducts, equalizers, and coequalizers, for a morphism we can form the following construction,