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Theorem List for Metamath Proof Explorer - 26701-26800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremeengtrkge 26701 The geometry structure for 𝔼↑𝑁 is a Euclidean geometry. (Contributed by Thierry Arnoux, 15-Mar-2019.)
(𝑁 ∈ ℕ → (EEG‘𝑁) ∈ TarskiGE)
 
PART 16  GRAPH THEORY



To give an overview of the definitions and terms used in the context of graph theory, a glossary is provided in the following, mainly according to definitions in [Bollobas] p. 1-8 or in [Diestel] p. 2-28. Although this glossary concentrates on undirected graphs, many of the concepts are also useful for directed graphs.

Basic concepts:

TermReferenceDefinitionRemarks
Vertex df-vtx 26711 A vertex of a graph 𝐺 is an element of the set of vertices (Vtx‘𝐺) of the graph 𝐺. The set of vertices (Vtx‘𝐺) (corresponding to V(G) in [Bollobas] p. 1) is usually the first component 𝑉 of the graph 𝐺 if it is represented by an ordered pair 𝑉, 𝐸 (see opvtxfv 26717), or the base set (Base‘𝐺) of the graph 𝐺 if it is represented as extensible structure (see basvtxval 26729).
Edge df-edg 26761 An edge of a graph 𝐺 is a nonempty set of vertices of the graph. It is said that these vertices are "joined" or "connected" by the edge, see [Bollobas] p. 1. The set of edges (Edg‘𝐺) (corresponding to E(G) in [Bollobas] p. 1) is usually the range ran 𝐸 of the second component 𝐸 of the graph 𝐺 if it is represented by an ordered pair 𝑉, 𝐸, or the range of the component (.ef‘𝐺) of the graph 𝐺 if it is represented as extensible structure.
Loop A loop in a graph 𝐺 is an edge which connects a single vertex with itself (or, according to [Bollobas] p. 7 "joins a vertex to itself"). In other words, a loop is an edge 𝑒 ∈ (Edg‘𝐺) which is a singleton consisting of a vertex 𝑣 ∈ (Vtx‘𝐺): 𝑒 = {𝑣}
Edge function resp. indexed edges df-iedg 26712 An edge function (or indexed set of edges) of a graph 𝐺 is a mapping of an arbitrary index set to nonempty sets of vertices of the graph. The edge function (iEdg‘𝐺) is usually the second component 𝐸 of the graph 𝐺 if it is represented by an ordered pair 𝑉, 𝐸 > (see opiedgfv 26720), or the component (.ef‘𝐺) of the graph 𝐺 if it is represented as extensible structure (see edgfiedgval 26730).
The set of edges of a graph 𝐺 is the range of its edge function: (Edg‘𝐺) = ran (iEdg‘𝐺), see edgval 26762.
Whereas the concept of plain edges is sufficient for simple hypergraphs, indexed edges are required for e.g. multigraphs in which the same vertices may be connected by more than one edge.

Basic kinds of graphs:

TermReferenceDefinitionRemarks
Undirected hypergraph df-uhgr 26771 a class 𝐺 with an edge function 𝐸 = (iEdg‘𝐺) which is a function into the power set of the vertices 𝑉 = (Vtx‘𝐺): ran 𝐸 ⊆ (𝒫 𝑉 ∖ {∅}). In this most general definition of a graph, an "edge" may connect three or more vertices with each other, see [Berge] p. 1.
In Wikipedia "Hypergraph", see https://en.wikipedia.org/wiki/Hypergraph 26771 (18-Jan-2020) such a hypergraph is called a "non-simple hypergraph", "multiple hypergraph" or "multi-hypergraphs". According to Wikipedia "Incidence structure", see https://en.wikipedia.org/wiki/Incidence_structure 26771 (18-Jan-2020) "Each hypergraph [...] can be regarded as an incidence structure in which the [vertices] play the role of "points", the corresponding family of [edges] plays the role of "lines" and the incidence relation is set membership".

Notice that by using (Edg‘𝐺) the (possibly more than one) edges connecting the same vertices could not be distinguished anymore. Therefore, this representation will only be used for undirected simple hypergraphs.
Undirected simple hypergraph df-ushgr 26772 a class 𝐺 with an edge function 𝐸 = (iEdg‘𝐺) which is a one-to-one function into the power set of the vertices 𝑉 = (Vtx‘𝐺): ran 𝐸 ⊆ (𝒫 𝑉 ∖ {∅}). See also Wikipedia "Hypergraph", https://en.wikipedia.org/wiki/Hypergraph 26772 (18-Jan-2020). This is how a "hypergraph" is defined in Section I.1 in [Bollobas] p. 7 or the definition in section 1.10 in [Diestel] p. 27. A simple hypergraph has at most one edge between the same vertices, hence a pseudograph needs not be a simple hypergraph.
According to [Berge] p. 1, "A simple hypergraph (or "Sperner family") is a hypergraph H = { E_1, E_2, ..., E_m } such that E_i C_ E_j => i = j". By this definition, a simple hypergraph cannot contain the edges E_1 = { v_1 , v_2 } and E_2 = { v_1, v_2, v_3 }, because E_1 C_ E_2, but 1 =/= 2.
Undirected loop-free hypergraph--- an undirected hypergraph without a loop, i.e. all edges connect at least two vertices.
Undirected pseudograph df-upgr 26795 a class 𝐺 with an edge function 𝐸 = (iEdg‘𝐺) which is a function into the set of (proper or not proper) unordered pairs of vertices 𝑉 = (Vtx‘𝐺). A proper unordered pair contains two different elements, a not proper unordered pair contains two times the same element, so it is a singleton (see preqsn 4786). This means a pseudograph may contain loops.
This definition corresponds to the definition of a "multigraph" in Section I.1 in [Bollobas] p. 7, "In a multigraph both multiple edges [joining two vertices] and multiple loops [joining a vertex to itself] are allowed", or in [Diestel] p. 28, "A multigraph is a pair (V,E) of disjoint sets (of vertices and edges) together with a map E -> V u. [V]^2 assigning to every edge either one or two vertices, its end(s).".
Undirected multigraph df-umgr 26796 a class 𝐺 with an edge function 𝐸 = (iEdg‘𝐺) which is a function into the set of (proper!) unordered pairs of vertices 𝑉 = (Vtx‘𝐺). This definition is according to Chartrand, Gary and Zhang, Ping (2012): "A First Course in Graph Theory.", Dover, ISBN 978-0-486-48368-9, section 1.4, p. 26: "A multigraph M consists of a finite nonempty set V of vertices and a set E of edges, where every two vertices of M are joined by a finite number of edges (possibly zero). If two or more edges join the same pair of (distinct) vertices, then these edges are called parallel edges."
A proper unordered pair contains two different elements, therefore a multigraph does not have loops.
Undirected simple pseudograph df-uspgr 26863 a class 𝐺 with an edge function 𝐸 = (iEdg‘𝐺) which is a one-to-one function into the set of (proper or not proper) unordered pairs of vertices 𝑉 = (Vtx‘𝐺). This means that there is at most one edge between two vertices, and at most one loop from a vertex to itself.
Undirected simple graph df-usgr 26864 a class 𝐺 with an edge function 𝐸 = (iEdg‘𝐺) which is a one-to-one function into the set of (proper!) unordered pairs of vertices 𝑉 = (Vtx‘𝐺). An ordered pair 𝑉, 𝐸 of two distinct sets 𝑉 (the vertices) and 𝐸 (the edges), the "usual" definition of a "graph", see, for example, the definition in section I.1 of [Bollobas] p. 1 or in section 1.1 of [Diestel] p. 2, can be identified with an undirected simple graph without loops by "indexing" the edges with themselves, see usgrausgrb 26882.
Finite graph df-fusgr 27027 a graph 𝐺 with a finite set of vertices 𝑉 = (Vtx‘𝐺). See definitions in [Bollobas] p. 1 or [Diestel] p. 2.
In simple graphs, the set of (indexed) edges (iEdg‘𝐺) (and therefore also the set of edges (Edg‘𝐺)) is finite if 𝑉 = (Vtx‘𝐺) is finite, see fusgrfis 27040. The number of edges is limited by (𝑛C2) (or "𝑛 choose 2") with 𝑛 = (♯‘𝑉), see fusgrmaxsize 27174. Analogously, the number of edges 𝐸 = (iEdg‘𝐺) of an undirected simple pseudograph (which may have loops) is limited by ((𝑛 + 1)C2). In pseudographs or multigraphs, however, 𝐸 can be infinite although 𝑉 is finite.
Graph of finite size--- a graph 𝐺 with a finite set 𝐸 = (iEdg‘𝐺), i.e. with a finite number of edges. A graph can be of finite size although its set of vertices is infinite (most of the vertices would not be connected by an edge).

Terms and properties of graphs:

TermReferenceDefinitionRemarks
Edge joining resp. connecting (two) vertices --- An edge 𝑒 ∈ (Edg‘𝐺) joins resp. connects the vertices v_1, v_2, ... v_n (𝑛 ∈ ℕ) if 𝑒 = { v_1, v_2, ... v_n }. If 𝑛 = 1, 𝑒 = { v_1 } is a loop, if 𝑛 = 2, 𝑒 = { v_1 , v_2 } is an edge as it is usually defined, see definition in Section I.1 in [Bollobas] p. 1.
(Two) Endvertices of an edge see definition in Section I.1 in [Bollobas] p. 1. If an edge 𝑒 ∈ (Edg‘𝐺) joins the vertices v_1, v_2, ... v_n (𝑛 ∈ ℕ), then the vertices v_1, v_2, ... v_n are called the endvertices of the edge 𝑒.
(Two) Adjacent vertices see definition in Section I.1 in [Bollobas] p. 1/2. The vertices v_1, v_2, ... v_n (𝑛 ∈ ℕ) are adjacent if there is an edge e = { v_1, v_2, ... v_n } joining these vertices. In this case, the vertices are incident with the edge e (see definition in Section I.1 in [Bollobas] p. 2) or connected by the edge e.
Edge ending at a vertex An edge 𝑒 ∈ (Edg‘𝐺) is ending at a vertex 𝑣 if the vertex is an endvertex of the edge: 𝑣𝑒. In other words, the vertex 𝑣 is incident with the edge 𝑒.
(Two) Adjacent edges The edges e_0, e_1, ... e_n (𝑛 ∈ ℕ) are adjacent if they have exactly one common endvertex. Generalization of definition in Section I.1 in [Bollobas] p. 2.
Order of a graph see definition in Section I.1 in [Bollobas] p. 3 The order of a graph 𝐺 is the number of vertices in the graph: (♯‘(Vtx‘𝐺)).
Size of a graph see definition in Section I.1 in [Bollobas] p. 3 The size of a graph 𝐺 is the number of edges in the graph: (♯‘(iEdg‘𝐺)). Or, for a simple graph 𝐺: (♯‘(Edg‘𝐺))).
Neighborhood of a vertex df-nbgr 27043 resp. definition in Section I.1 in [Bollobas] p. 3 A vertex connected with a vertex 𝑣 by an edge is called a neighbor of the vertex 𝑣. The set of neighbors of a vertex 𝑣 is called the neighborhood (or open neighborhood) of the vertex 𝑣. The closed neighborhood is the union of the (open) neighborhood of the vertex 𝑣 with {𝑣}.
Degree of a vertex df-vtxdg 27176 The degree of a vertex is the number of the edges ending at this vertex. In a simple graph, the degree of a vertex is the number of neighbors of this vertex, see definition in Section I.1 in [Bollobas] p. 3
Isolated vertex usgrvd0nedg 27243 A vertex is called isolated if it is not an endvertex of any edge, thus having degree 0.
Universal vertex df-uvtx 27096 A vertex is called universal if it is connected with every other vertex of the graph by an edge, thus having degree ((♯‘(Vtx‘𝐺)) − ).

Special kinds of graphs:

TermReferenceDefinitionRemarks
Complete graph df-cplgr 27121 A graph is called complete if each pair of vertices is connected by an edge. The size of a complete undirected simple graph of order 𝑛 is (𝑛C2) (or "𝑛 choose 2"), see cusgrsize 27164.
Empty graph uhgr0e 26784 A graph is called empty if it has no edges.
Null graph uhgr0 26786 and uhgr0vb 26785 A graph is called a null graph if it has no vertices (and therefore also no edges).
Trivial graph usgr1v 26966 A graph is called the "trivial graph" if it has only one vertex and no edges.
Connected graph df-conngr 27894 resp. definition in Section I.1 in [Bollobas] p. 6 A graph is called connected if for each pair of vertices there is a path between these vertices.


For the terms "Path", "Walk", "Trail", "Circuit", "Cycle" see the remarks below and the definitions in Section I.1 in [Bollobas] p. 4-5.
 
16.1  Vertices and edges

In the following, the vertices and (indexed) edges for an arbitrary class 𝐺 (called "graph" in the following) are defined and examined. The main result of this section is to show that the set of vertices (Vtx‘𝐺) of a graph 𝐺 is the first component 𝑉 of the graph 𝐺 if it is represented by an ordered pair 𝑉, 𝐸 (see opvtxfv 26717), or the base set (Base‘𝐺) of the graph 𝐺 if it is represented as extensible structure (see basvtxval 26729), and that the set of indexed edges resp. the edge function (iEdg‘𝐺) is the second component 𝐸 of the graph 𝐺 if it is represented by an ordered pair 𝑉, 𝐸 (see opiedgfv 26720), or the component (.ef‘𝐺) of the graph 𝐺 if it is represented as extensible structure (see edgfiedgval 26730). Finally, it is shown that the set of edges of a graph 𝐺 is the range of its edge function: (Edg‘𝐺) = ran (iEdg‘𝐺), see edgval 26762.

Usually, a graph 𝐺 is a set. If 𝐺 is a proper class, however, it represents the null graph (without vertices and edges), because (Vtx‘𝐺) = ∅ and (iEdg‘𝐺) = ∅ holds, see vtxvalprc 26758 and iedgvalprc 26759.

Up to the end of this section, the edges need not be related to the vertices. Once undirected hypergraphs are defined (see df-uhgr 26771), the edges become nonempty sets of vertices, and by this obtain their meaning as "connectors" of vertices.

 
16.1.1  The edge function extractor for extensible structures
 
Syntaxcedgf 26702 Extend class notation with an edge function.
class .ef
 
Definitiondf-edgf 26703 Define the edge function (indexed edges) of a graph. (Contributed by AV, 18-Jan-2020.)
.ef = Slot 18
 
Theoremedgfid 26704 Utility theorem: index-independent form of df-edgf 26703. (Contributed by AV, 16-Nov-2021.)
.ef = Slot (.ef‘ndx)
 
Theoremedgfndxnn 26705 The index value of the edge function extractor is a positive integer. This property should be ensured for every concrete coding because otherwise it could not be used in an extensible structure (slots must be positive integers). (Contributed by AV, 21-Sep-2020.)
(.ef‘ndx) ∈ ℕ
 
Theoremedgfndxid 26706 The value of the edge function extractor is the value of the corresponding slot of the structure. (Contributed by AV, 21-Sep-2020.)
(𝐺𝑉 → (.ef‘𝐺) = (𝐺‘(.ef‘ndx)))
 
Theorembaseltedgf 26707 The index value of the Base slot is less than the index value of the .ef slot. (Contributed by AV, 21-Sep-2020.)
(Base‘ndx) < (.ef‘ndx)
 
Theoremslotsbaseefdif 26708 The slots Base and .ef are different. (Contributed by AV, 21-Sep-2020.)
(Base‘ndx) ≠ (.ef‘ndx)
 
16.1.2  Vertices and indexed edges

The key concepts in graph theory are vertices and edges. In general, a graph "consists" (at least) of two sets: the set of vertices and the set of edges. The edges "connect" vertices. The meaning of "connect" is different for different kinds of graphs (directed/undirected graphs, hyper-/pseudo-/ multi-/simple graphs, etc.). The simplest way to represent a graph (of any kind) is to define a graph as "an ordered pair of disjoint sets (V, E)" (see section I.1 in [Bollobas] p. 1), or in the notation of Metamath: 𝑉, 𝐸.

Another way is to regard a graph as a mathematical structure, which consistes at least of a set (of vertices) and a relation between the vertices (edge function), but which can be enhanced by additional features (see Wikipedia "Mathematical structure", 24-Sep-2020, https://en.wikipedia.org/wiki/Mathematical_structure): "In mathematics, a structure is a set endowed with some additional features on the set (e.g., operation, relation, metric, topology). Often, the additional features are attached or related to the set, so as to provide it with some additional meaning or significance.". Such structures are provided as "extensible structures" in Metamath, see df-struct 16475.

To allow for expressing and proving most of the theorems for graphs independently from their representation, the functions Vtx and iEdg are defined (see df-vtx 26711 and df-iedg 26712), which provide the vertices resp. (indexed) edges of an arbitrary class 𝐺 which represents a graph: (Vtx‘𝐺) resp. (iEdg‘𝐺). In literature, these functions are often denoted also by "V" and "E", see section I.1 in [Bollobas] p. 1 ("If G is a graph, then V = V(G) is the vertex set of G, and E = E(G) is the edge set.") or section 1.1 in [Diestel] p. 2 ("The vertex set of graph G is referred to as V(G), its edge set as E(G).").

Instead of providing edges themselves, iEdg is intended to provide a function as mapping of "indices" (the domain of the function) to the edges (therefore called "set of indexed edges"), which allows for hyper-/pseudo-/multigraphs with more than one edge between two (or more) vertices. For example, e1 = e(1) = { a, b } and e2 = e(2) = { a, b } are two different edges connecting the same two vertices a and b (in a pseudograph). In section 1.10 of [Diestel] p. 28, the edge function is defined differently: as "map E -> V u. [V]^2 assigning to every edge either one or two vertices, its end.". Here, the domain is the set of abstract edges: for two different edges e1 and e2 connecting the same two vertices a and b, we would have e(e1) = e(e2) = { a, b }. Since the set of abstract edges can be chosen as index set, these definitions are equivalent.

The result of these functions are as expected: for a graph represented as ordered pair (𝐺 ∈ (V × V)), the set of vertices is (Vtx‘𝐺) = (1st𝐺) (see opvtxval 26716) and the set of (indexed) edges is (iEdg‘𝐺) = (2nd𝐺) (see opiedgval 26719), or if 𝐺 is given as ordered pair 𝐺 = ⟨𝑉, 𝐸, the set of vertices is (Vtx‘𝐺) = 𝑉 (see opvtxfv 26717) and the set of (indexed) edges is (iEdg‘𝐺) = 𝐸 (see opiedgfv 26720).

And for a graph represented as extensible structure (𝐺 Struct ⟨(Base‘ndx), (.ef‘ndx)⟩), the set of vertices is (Vtx‘𝐺) = (Base‘𝐺) (see funvtxval 26731) and the set of (indexed) edges is (iEdg‘𝐺) = (.ef‘𝐺) (see funiedgval 26732), or if 𝐺 is given in its simplest form as extensible structure with two slots (𝐺 = {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩}), the set of vertices is (Vtx‘𝐺) = 𝑉 (see struct2grvtx 26740) and the set of (indexed) edges is (iEdg‘𝐺) = 𝐸 (see struct2griedg 26741).

These two representations are convertible, see graop 26742 and grastruct 26743: If 𝐺 is a graph (for example 𝐺 = ⟨𝑉, 𝐸), then 𝐻 = {⟨(Base‘ndx), (Vtx‘𝐺)⟩, ⟨(.ef‘ndx), (iEdg‘𝐺)⟩} represents essentially the same graph, and if 𝐺 is a graph (for example 𝐺 = {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩}), then 𝐻 = ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ represents essentially the same graph. In both cases, (Vtx‘𝐺) = (Vtx‘𝐻) and (iEdg‘𝐺) = (iEdg‘𝐻) hold. Theorems gropd 26744 and gropeld 26746 show that if any representation of a graph with vertices 𝑉 and edges 𝐸 has a certain property, then the ordered pair 𝑉, 𝐸 of the set of vertices and the set of edges (which is such a representation of a graph with vertices 𝑉 and edges 𝐸) has this property. Analogously, theorems grstructd 26745 and grstructeld 26747 show that if any representation of a graph with vertices 𝑉 and edges 𝐸 has a certain property, then any extensible structure with base set 𝑉 and value 𝐸 in the slot for edge functions (which is also such a representation of a graph with vertices 𝑉 and edges 𝐸) has this property.

Besides the usual way to represent graphs without edges (consisting of unconnected vertices only), which would be 𝐺 = ⟨𝑉, ∅⟩ or 𝐺 = {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), ∅⟩}, a structure without a slot for edges can be used: 𝐺 = {⟨(Base‘ndx), 𝑉⟩}, see snstrvtxval 26750 and snstriedgval 26751. Analogously, the empty set can be used to represent the null graph, see vtxval0 26752 and iedgval0 26753, which can also be represented by 𝐺 = ⟨∅, ∅⟩ or 𝐺 = {⟨(Base‘ndx), ∅⟩, ⟨(.ef‘ndx), ∅⟩}. Even proper classes can be used to represent the null graph, see vtxvalprc 26758 and iedgvalprc 26759.

Other classes should not be used to represent graphs, because there could be a degenerate behavior of the vertex set and (indexed) edge functions, see vtxvalsnop 26754 resp. iedgvalsnop 26755, and vtxval3sn 26756 resp. iedgval3sn 26757. Avoid directly depending on this detail so that theorems will not depend on the Kuratowski construction of ordered pairs, see also the comment for df-op 4566.

 
16.1.2.1  Definitions and basic properties
 
Syntaxcvtx 26709 Extend class notation with the vertices of "graphs".
class Vtx
 
Syntaxciedg 26710 Extend class notation with the indexed edges of "graphs".
class iEdg
 
Definitiondf-vtx 26711 Define the function mapping a graph to the set of its vertices. This definition is very general: It defines the set of vertices for any ordered pair as its first component, and for any other class as its "base set". It is meaningful, however, only if the ordered pair represents a graph resp. the class is an extensible structure representing a graph. (Contributed by AV, 9-Jan-2020.) (Revised by AV, 20-Sep-2020.)
Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st𝑔), (Base‘𝑔)))
 
Definitiondf-iedg 26712 Define the function mapping a graph to its indexed edges. This definition is very general: It defines the indexed edges for any ordered pair as its second component, and for any other class as its "edge function". It is meaningful, however, only if the ordered pair represents a graph resp. the class is an extensible structure (containing a slot for "edge functions") representing a graph. (Contributed by AV, 20-Sep-2020.)
iEdg = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (2nd𝑔), (.ef‘𝑔)))
 
Theoremvtxval 26713 The set of vertices of a graph. (Contributed by AV, 9-Jan-2020.) (Revised by AV, 21-Sep-2020.)
(Vtx‘𝐺) = if(𝐺 ∈ (V × V), (1st𝐺), (Base‘𝐺))
 
Theoremiedgval 26714 The set of indexed edges of a graph. (Contributed by AV, 21-Sep-2020.)
(iEdg‘𝐺) = if(𝐺 ∈ (V × V), (2nd𝐺), (.ef‘𝐺))
 
Theorem1vgrex 26715 A graph with at least one vertex is a set. (Contributed by AV, 2-Mar-2021.)
𝑉 = (Vtx‘𝐺)       (𝑁𝑉𝐺 ∈ V)
 
16.1.2.2  The vertices and edges of a graph represented as ordered pair
 
Theoremopvtxval 26716 The set of vertices of a graph represented as an ordered pair of vertices and indexed edges. (Contributed by AV, 9-Jan-2020.) (Revised by AV, 21-Sep-2020.)
(𝐺 ∈ (V × V) → (Vtx‘𝐺) = (1st𝐺))
 
Theoremopvtxfv 26717 The set of vertices of a graph represented as an ordered pair of vertices and indexed edges as function value. (Contributed by AV, 21-Sep-2020.)
((𝑉𝑋𝐸𝑌) → (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉)
 
Theoremopvtxov 26718 The set of vertices of a graph represented as an ordered pair of vertices and indexed edges as operation value. (Contributed by AV, 21-Sep-2020.)
((𝑉𝑋𝐸𝑌) → (𝑉Vtx𝐸) = 𝑉)
 
Theoremopiedgval 26719 The set of indexed edges of a graph represented as an ordered pair of vertices and indexed edges. (Contributed by AV, 21-Sep-2020.)
(𝐺 ∈ (V × V) → (iEdg‘𝐺) = (2nd𝐺))
 
Theoremopiedgfv 26720 The set of indexed edges of a graph represented as an ordered pair of vertices and indexed edges as function value. (Contributed by AV, 21-Sep-2020.)
((𝑉𝑋𝐸𝑌) → (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)
 
Theoremopiedgov 26721 The set of indexed edges of a graph represented as an ordered pair of vertices and indexed edges as operation value. (Contributed by AV, 21-Sep-2020.)
((𝑉𝑋𝐸𝑌) → (𝑉iEdg𝐸) = 𝐸)
 
Theoremopvtxfvi 26722 The set of vertices of a graph represented as an ordered pair of vertices and indexed edges as function value. (Contributed by AV, 4-Mar-2021.)
𝑉 ∈ V    &   𝐸 ∈ V       (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉
 
Theoremopiedgfvi 26723 The set of indexed edges of a graph represented as an ordered pair of vertices and indexed edges as function value. (Contributed by AV, 4-Mar-2021.)
𝑉 ∈ V    &   𝐸 ∈ V       (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸
 
16.1.2.3  The vertices and edges of a graph represented as extensible structure
 
Theoremfunvtxdmge2val 26724 The set of vertices of an extensible structure with (at least) two slots. (Contributed by AV, 12-Oct-2020.) (Revised by AV, 7-Jun-2021.) (Revised by AV, 12-Nov-2021.)
((Fun (𝐺 ∖ {∅}) ∧ 2 ≤ (♯‘dom 𝐺)) → (Vtx‘𝐺) = (Base‘𝐺))
 
Theoremfuniedgdmge2val 26725 The set of indexed edges of an extensible structure with (at least) two slots. (Contributed by AV, 12-Oct-2020.) (Revised by AV, 7-Jun-2021.) (Revised by AV, 12-Nov-2021.)
((Fun (𝐺 ∖ {∅}) ∧ 2 ≤ (♯‘dom 𝐺)) → (iEdg‘𝐺) = (.ef‘𝐺))
 
Theoremfunvtxdm2val 26726 The set of vertices of an extensible structure with (at least) two slots. (Contributed by AV, 22-Sep-2020.) (Revised by AV, 7-Jun-2021.) (Revised by AV, 12-Nov-2021.)
𝐴 ∈ V    &   𝐵 ∈ V       ((Fun (𝐺 ∖ {∅}) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵} ⊆ dom 𝐺) → (Vtx‘𝐺) = (Base‘𝐺))
 
Theoremfuniedgdm2val 26727 The set of indexed edges of an extensible structure with (at least) two slots. (Contributed by AV, 22-Sep-2020.) (Revised by AV, 7-Jun-2021.) (Revised by AV, 12-Nov-2021.)
𝐴 ∈ V    &   𝐵 ∈ V       ((Fun (𝐺 ∖ {∅}) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵} ⊆ dom 𝐺) → (iEdg‘𝐺) = (.ef‘𝐺))
 
Theoremfunvtxval0 26728 The set of vertices of an extensible structure with a base set and (at least) another slot. (Contributed by AV, 22-Sep-2020.) (Revised by AV, 7-Jun-2021.) (Revised by AV, 12-Nov-2021.)
𝑆 ∈ V       ((Fun (𝐺 ∖ {∅}) ∧ 𝑆 ≠ (Base‘ndx) ∧ {(Base‘ndx), 𝑆} ⊆ dom 𝐺) → (Vtx‘𝐺) = (Base‘𝐺))
 
Theorembasvtxval 26729 The set of vertices of a graph represented as an extensible structure with the set of vertices as base set. (Contributed by AV, 14-Oct-2020.) (Revised by AV, 12-Nov-2021.)
(𝜑𝐺 Struct 𝑋)    &   (𝜑 → 2 ≤ (♯‘dom 𝐺))    &   (𝜑𝑉𝑌)    &   (𝜑 → ⟨(Base‘ndx), 𝑉⟩ ∈ 𝐺)       (𝜑 → (Vtx‘𝐺) = 𝑉)
 
Theoremedgfiedgval 26730 The set of indexed edges of a graph represented as an extensible structure with the indexed edges in the slot for edge functions. (Contributed by AV, 14-Oct-2020.) (Revised by AV, 12-Nov-2021.)
(𝜑𝐺 Struct 𝑋)    &   (𝜑 → 2 ≤ (♯‘dom 𝐺))    &   (𝜑𝐸𝑌)    &   (𝜑 → ⟨(.ef‘ndx), 𝐸⟩ ∈ 𝐺)       (𝜑 → (iEdg‘𝐺) = 𝐸)
 
Theoremfunvtxval 26731 The set of vertices of a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 22-Sep-2020.) (Revised by AV, 7-Jun-2021.) (Revised by AV, 12-Nov-2021.)
((Fun (𝐺 ∖ {∅}) ∧ {(Base‘ndx), (.ef‘ndx)} ⊆ dom 𝐺) → (Vtx‘𝐺) = (Base‘𝐺))
 
Theoremfuniedgval 26732 The set of indexed edges of a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 21-Sep-2020.) (Revised by AV, 7-Jun-2021.) (Revised by AV, 12-Nov-2021.)
((Fun (𝐺 ∖ {∅}) ∧ {(Base‘ndx), (.ef‘ndx)} ⊆ dom 𝐺) → (iEdg‘𝐺) = (.ef‘𝐺))
 
Theoremstructvtxvallem 26733 Lemma for structvtxval 26734 and structiedg0val 26735. (Contributed by AV, 23-Sep-2020.) (Revised by AV, 12-Nov-2021.)
𝑆 ∈ ℕ    &   (Base‘ndx) < 𝑆    &   𝐺 = {⟨(Base‘ndx), 𝑉⟩, ⟨𝑆, 𝐸⟩}       ((𝑉𝑋𝐸𝑌) → 2 ≤ (♯‘dom 𝐺))
 
Theoremstructvtxval 26734 The set of vertices of an extensible structure with a base set and another slot. (Contributed by AV, 23-Sep-2020.) (Proof shortened by AV, 12-Nov-2021.)
𝑆 ∈ ℕ    &   (Base‘ndx) < 𝑆    &   𝐺 = {⟨(Base‘ndx), 𝑉⟩, ⟨𝑆, 𝐸⟩}       ((𝑉𝑋𝐸𝑌) → (Vtx‘𝐺) = 𝑉)
 
Theoremstructiedg0val 26735 The set of indexed edges of an extensible structure with a base set and another slot not being the slot for edge functions is empty. (Contributed by AV, 23-Sep-2020.) (Proof shortened by AV, 12-Nov-2021.)
𝑆 ∈ ℕ    &   (Base‘ndx) < 𝑆    &   𝐺 = {⟨(Base‘ndx), 𝑉⟩, ⟨𝑆, 𝐸⟩}       ((𝑉𝑋𝐸𝑌𝑆 ≠ (.ef‘ndx)) → (iEdg‘𝐺) = ∅)
 
Theoremstructgrssvtxlem 26736 Lemma for structgrssvtx 26737 and structgrssiedg 26738. (Contributed by AV, 14-Oct-2020.) (Proof shortened by AV, 12-Nov-2021.)
(𝜑𝐺 Struct 𝑋)    &   (𝜑𝑉𝑌)    &   (𝜑𝐸𝑍)    &   (𝜑 → {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩} ⊆ 𝐺)       (𝜑 → 2 ≤ (♯‘dom 𝐺))
 
Theoremstructgrssvtx 26737 The set of vertices of a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 14-Oct-2020.) (Proof shortened by AV, 12-Nov-2021.)
(𝜑𝐺 Struct 𝑋)    &   (𝜑𝑉𝑌)    &   (𝜑𝐸𝑍)    &   (𝜑 → {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩} ⊆ 𝐺)       (𝜑 → (Vtx‘𝐺) = 𝑉)
 
Theoremstructgrssiedg 26738 The set of indexed edges of a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 14-Oct-2020.) (Proof shortened by AV, 12-Nov-2021.)
(𝜑𝐺 Struct 𝑋)    &   (𝜑𝑉𝑌)    &   (𝜑𝐸𝑍)    &   (𝜑 → {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩} ⊆ 𝐺)       (𝜑 → (iEdg‘𝐺) = 𝐸)
 
Theoremstruct2grstr 26739 A graph represented as an extensible structure with vertices as base set and indexed edges is actually an extensible structure. (Contributed by AV, 23-Nov-2020.)
𝐺 = {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩}       𝐺 Struct ⟨(Base‘ndx), (.ef‘ndx)⟩
 
Theoremstruct2grvtx 26740 The set of vertices of a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 23-Sep-2020.)
𝐺 = {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩}       ((𝑉𝑋𝐸𝑌) → (Vtx‘𝐺) = 𝑉)
 
Theoremstruct2griedg 26741 The set of indexed edges of a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 23-Sep-2020.) (Proof shortened by AV, 12-Nov-2021.)
𝐺 = {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩}       ((𝑉𝑋𝐸𝑌) → (iEdg‘𝐺) = 𝐸)
 
Theoremgraop 26742 Any representation of a graph 𝐺 (especially as extensible structure 𝐺 = {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩}) is convertible in a representation of the graph as ordered pair. (Contributed by AV, 7-Oct-2020.)
𝐻 = ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩       ((Vtx‘𝐺) = (Vtx‘𝐻) ∧ (iEdg‘𝐺) = (iEdg‘𝐻))
 
Theoremgrastruct 26743 Any representation of a graph 𝐺 (especially as ordered pair 𝐺 = ⟨𝑉, 𝐸) is convertible in a representation of the graph as extensible structure. (Contributed by AV, 8-Oct-2020.)
𝐻 = {⟨(Base‘ndx), (Vtx‘𝐺)⟩, ⟨(.ef‘ndx), (iEdg‘𝐺)⟩}       ((Vtx‘𝐺) = (Vtx‘𝐻) ∧ (iEdg‘𝐺) = (iEdg‘𝐻))
 
Theoremgropd 26744* If any representation of a graph with vertices 𝑉 and edges 𝐸 has a certain property 𝜓, then the ordered pair 𝑉, 𝐸 of the set of vertices and the set of edges (which is such a representation of a graph with vertices 𝑉 and edges 𝐸) has this property. (Contributed by AV, 11-Oct-2020.)
(𝜑 → ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) → 𝜓))    &   (𝜑𝑉𝑈)    &   (𝜑𝐸𝑊)       (𝜑[𝑉, 𝐸⟩ / 𝑔]𝜓)
 
Theoremgrstructd 26745* If any representation of a graph with vertices 𝑉 and edges 𝐸 has a certain property 𝜓, then any structure with base set 𝑉 and value 𝐸 in the slot for edge functions (which is such a representation of a graph with vertices 𝑉 and edges 𝐸) has this property. (Contributed by AV, 12-Oct-2020.) (Revised by AV, 9-Jun-2021.)
(𝜑 → ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) → 𝜓))    &   (𝜑𝑉𝑈)    &   (𝜑𝐸𝑊)    &   (𝜑𝑆𝑋)    &   (𝜑 → Fun (𝑆 ∖ {∅}))    &   (𝜑 → 2 ≤ (♯‘dom 𝑆))    &   (𝜑 → (Base‘𝑆) = 𝑉)    &   (𝜑 → (.ef‘𝑆) = 𝐸)       (𝜑[𝑆 / 𝑔]𝜓)
 
Theoremgropeld 26746* If any representation of a graph with vertices 𝑉 and edges 𝐸 is an element of an arbitrary class 𝐶, then the ordered pair 𝑉, 𝐸 of the set of vertices and the set of edges (which is such a representation of a graph with vertices 𝑉 and edges 𝐸) is an element of this class 𝐶. (Contributed by AV, 11-Oct-2020.)
(𝜑 → ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) → 𝑔𝐶))    &   (𝜑𝑉𝑈)    &   (𝜑𝐸𝑊)       (𝜑 → ⟨𝑉, 𝐸⟩ ∈ 𝐶)
 
Theoremgrstructeld 26747* If any representation of a graph with vertices 𝑉 and edges 𝐸 is an element of an arbitrary class 𝐶, then any structure with base set 𝑉 and value 𝐸 in the slot for edge functions (which is such a representation of a graph with vertices 𝑉 and edges 𝐸) is an element of this class 𝐶. (Contributed by AV, 12-Oct-2020.) (Revised by AV, 9-Jun-2021.)
(𝜑 → ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) → 𝑔𝐶))    &   (𝜑𝑉𝑈)    &   (𝜑𝐸𝑊)    &   (𝜑𝑆𝑋)    &   (𝜑 → Fun (𝑆 ∖ {∅}))    &   (𝜑 → 2 ≤ (♯‘dom 𝑆))    &   (𝜑 → (Base‘𝑆) = 𝑉)    &   (𝜑 → (.ef‘𝑆) = 𝐸)       (𝜑𝑆𝐶)
 
Theoremsetsvtx 26748 The vertices of a structure with a base set and an inserted resp. replaced slot for the edge function. (Contributed by AV, 18-Jan-2020.) (Revised by AV, 16-Nov-2021.)
𝐼 = (.ef‘ndx)    &   (𝜑𝐺 Struct 𝑋)    &   (𝜑 → (Base‘ndx) ∈ dom 𝐺)    &   (𝜑𝐸𝑊)       (𝜑 → (Vtx‘(𝐺 sSet ⟨𝐼, 𝐸⟩)) = (Base‘𝐺))
 
Theoremsetsiedg 26749 The (indexed) edges of a structure with a base set and an inserted resp. replaced slot for the edge function. (Contributed by AV, 7-Jun-2021.) (Revised by AV, 16-Nov-2021.)
𝐼 = (.ef‘ndx)    &   (𝜑𝐺 Struct 𝑋)    &   (𝜑 → (Base‘ndx) ∈ dom 𝐺)    &   (𝜑𝐸𝑊)       (𝜑 → (iEdg‘(𝐺 sSet ⟨𝐼, 𝐸⟩)) = 𝐸)
 
16.1.2.4  Representations of graphs without edges
 
Theoremsnstrvtxval 26750 The set of vertices of a graph without edges represented as an extensible structure with vertices as base set and no indexed edges. See vtxvalsnop 26754 for the (degenerate) case where 𝑉 = (Base‘ndx). (Contributed by AV, 23-Sep-2020.)
𝑉 ∈ V    &   𝐺 = {⟨(Base‘ndx), 𝑉⟩}       (𝑉 ≠ (Base‘ndx) → (Vtx‘𝐺) = 𝑉)
 
Theoremsnstriedgval 26751 The set of indexed edges of a graph without edges represented as an extensible structure with vertices as base set and no indexed edges. See iedgvalsnop 26755 for the (degenerate) case where 𝑉 = (Base‘ndx). (Contributed by AV, 24-Sep-2020.)
𝑉 ∈ V    &   𝐺 = {⟨(Base‘ndx), 𝑉⟩}       (𝑉 ≠ (Base‘ndx) → (iEdg‘𝐺) = ∅)
 
16.1.2.5  Degenerated cases of representations of graphs
 
Theoremvtxval0 26752 Degenerated case 1 for vertices: The set of vertices of the empty set is the empty set. (Contributed by AV, 24-Sep-2020.)
(Vtx‘∅) = ∅
 
Theoremiedgval0 26753 Degenerated case 1 for edges: The set of indexed edges of the empty set is the empty set. (Contributed by AV, 24-Sep-2020.)
(iEdg‘∅) = ∅
 
Theoremvtxvalsnop 26754 Degenerated case 2 for vertices: The set of vertices of a singleton containing an ordered pair with equal components is the singleton containing the component. (Contributed by AV, 24-Sep-2020.) (Proof shortened by AV, 15-Jul-2022.) (Avoid depending on this detail.)
𝐵 ∈ V    &   𝐺 = {⟨𝐵, 𝐵⟩}       (Vtx‘𝐺) = {𝐵}
 
Theoremiedgvalsnop 26755 Degenerated case 2 for edges: The set of indexed edges of a singleton containing an ordered pair with equal components is the singleton containing the component. (Contributed by AV, 24-Sep-2020.) (Proof shortened by AV, 15-Jul-2022.) (Avoid depending on this detail.)
𝐵 ∈ V    &   𝐺 = {⟨𝐵, 𝐵⟩}       (iEdg‘𝐺) = {𝐵}
 
Theoremvtxval3sn 26756 Degenerated case 3 for vertices: The set of vertices of a singleton containing a singleton containing a singleton is the innermost singleton. (Contributed by AV, 24-Sep-2020.) (Avoid depending on this detail.)
𝐴 ∈ V       (Vtx‘{{{𝐴}}}) = {𝐴}
 
Theoremiedgval3sn 26757 Degenerated case 3 for edges: The set of indexed edges of a singleton containing a singleton containing a singleton is the innermost singleton. (Contributed by AV, 24-Sep-2020.) (Avoid depending on this detail.)
𝐴 ∈ V       (iEdg‘{{{𝐴}}}) = {𝐴}
 
Theoremvtxvalprc 26758 Degenerated case 4 for vertices: The set of vertices of a proper class is the empty set. (Contributed by AV, 12-Oct-2020.)
(𝐶 ∉ V → (Vtx‘𝐶) = ∅)
 
Theoremiedgvalprc 26759 Degenerated case 4 for edges: The set of indexed edges of a proper class is the empty set. (Contributed by AV, 12-Oct-2020.)
(𝐶 ∉ V → (iEdg‘𝐶) = ∅)
 
16.1.3  Edges as range of the edge function
 
Syntaxcedg 26760 Extend class notation with the set of edges (of an undirected simple (hyper-/pseudo-)graph).
class Edg
 
Definitiondf-edg 26761 Define the class of edges of a graph, see also definition "E = E(G)" in section I.1 of [Bollobas] p. 1. This definition is very general: It defines edges of a class as the range of its edge function (which does not even need to be a function). Therefore, this definition could also be used for hypergraphs, pseudographs and multigraphs. In these cases, however, the (possibly more than one) edges connecting the same vertices could not be distinguished anymore. In some cases, this is no problem, so theorems with Edg are meaningful nevertheless (e.g., edguhgr 26842). Usually, however, this definition is used only for undirected simple (hyper-/pseudo-)graphs (with or without loops). (Contributed by AV, 1-Jan-2020.) (Revised by AV, 13-Oct-2020.)
Edg = (𝑔 ∈ V ↦ ran (iEdg‘𝑔))
 
Theoremedgval 26762 The edges of a graph. (Contributed by AV, 1-Jan-2020.) (Revised by AV, 13-Oct-2020.) (Revised by AV, 8-Dec-2021.)
(Edg‘𝐺) = ran (iEdg‘𝐺)
 
Theoremiedgedg 26763 An indexed edge is an edge. (Contributed by AV, 19-Dec-2021.)
𝐸 = (iEdg‘𝐺)       ((Fun 𝐸𝐼 ∈ dom 𝐸) → (𝐸𝐼) ∈ (Edg‘𝐺))
 
Theoremedgopval 26764 The edges of a graph represented as ordered pair. (Contributed by AV, 1-Jan-2020.) (Revised by AV, 13-Oct-2020.)
((𝑉𝑊𝐸𝑋) → (Edg‘⟨𝑉, 𝐸⟩) = ran 𝐸)
 
Theoremedgov 26765 The edges of a graph represented as ordered pair, shown as operation value. Although a little less intuitive, this representation is often used because it is shorter than the representation as function value of a graph given as ordered pair, see edgopval 26764. The representation ran 𝐸 for the set of edges is even shorter, though. (Contributed by AV, 2-Jan-2020.) (Revised by AV, 13-Oct-2020.)
((𝑉𝑊𝐸𝑋) → (𝑉Edg𝐸) = ran 𝐸)
 
Theoremedgstruct 26766 The edges of a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 13-Oct-2020.)
𝐺 = {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩}       ((𝑉𝑊𝐸𝑋) → (Edg‘𝐺) = ran 𝐸)
 
Theoremedgiedgb 26767* A set is an edge iff it is an indexed edge. (Contributed by AV, 17-Oct-2020.) (Revised by AV, 8-Dec-2021.)
𝐼 = (iEdg‘𝐺)       (Fun 𝐼 → (𝐸 ∈ (Edg‘𝐺) ↔ ∃𝑥 ∈ dom 𝐼 𝐸 = (𝐼𝑥)))
 
Theoremedg0iedg0 26768 There is no edge in a graph iff its edge function is empty. (Contributed by AV, 15-Dec-2020.) (Revised by AV, 8-Dec-2021.)
𝐼 = (iEdg‘𝐺)    &   𝐸 = (Edg‘𝐺)       (Fun 𝐼 → (𝐸 = ∅ ↔ 𝐼 = ∅))
 
16.2  Undirected graphs

For undirected graphs, we will have the following hierarchy/taxonomy:

* Undirected Hypergraph: UHGraph

* Undirected loop-free graphs: ULFGraph (not defined formally yet)

* Undirected simple Hypergraph: USHGraph => USHGraph ⊆ UHGraph (ushgruhgr 26782)

* Undirected Pseudograph: UPGraph => UPGraph ⊆ UHGraph (upgruhgr 26815)

* Undirected loop-free hypergraph: ULFHGraph (not defined formally yet) => ULFHGraph ⊆ UHGraph and ULFHGraph ULFGraph

* Undirected loop-free simple hypergraph: ULFSHGraph (not defined formally yet) => ULFSHGraph ⊆ USHGraph and ULFSHGraph ULFHGraph

* Undirected simple Pseudograph: USPGraph => USPGraph ⊆ UPGraph (uspgrupgr 26889) and USPGraph ⊆ USHGraph (uspgrushgr 26888), see also uspgrupgrushgr 26890

* Undirected Muligraph: UMGraph => UMGraph ⊆ UPGraph (umgrupgr 26816) and UMGraph ⊆ ULFHGraph (umgrislfupgr 26836)

* Undirected simple Graph: USGraph => USGraph ⊆ USPGraph (usgruspgr 26891) and USGraph ⊆ UMGraph (usgrumgr 26892) and USGraph ⊆ ULFSHGraph (usgrislfuspgr 26897) see also usgrumgruspgr 26893

 
16.2.1  Undirected hypergraphs
 
Syntaxcuhgr 26769 Extend class notation with undirected hypergraphs.
class UHGraph
 
Syntaxcushgr 26770 Extend class notation with undirected simple hypergraphs.
class USHGraph
 
Definitiondf-uhgr 26771* Define the class of all undirected hypergraphs. An undirected hypergraph consists of a set 𝑣 (of "vertices") and a function 𝑒 (representing indexed "edges") into the power set of this set (the empty set excluded). (Contributed by Alexander van der Vekens, 26-Dec-2017.) (Revised by AV, 8-Oct-2020.)
UHGraph = {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶(𝒫 𝑣 ∖ {∅})}
 
Definitiondf-ushgr 26772* Define the class of all undirected simple hypergraphs. An undirected simple hypergraph is a special (non-simple, multiple, multi-) hypergraph for which the edge function 𝑒 is an injective (one-to-one) function into subsets of the set of vertices 𝑣, representing the (one or more) vertices incident to the edge. This definition corresponds to the definition of hypergraphs in section I.1 of [Bollobas] p. 7 (except that the empty set seems to be allowed to be an "edge") or section 1.10 of [Diestel] p. 27, where "E is a subset of [...] the power set of V, that is the set of all subsets of V" resp. "the elements of E are nonempty subsets (of any cardinality) of V". (Contributed by AV, 19-Jan-2020.) (Revised by AV, 8-Oct-2020.)
USHGraph = {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒1-1→(𝒫 𝑣 ∖ {∅})}
 
Theoremisuhgr 26773 The predicate "is an undirected hypergraph." (Contributed by Alexander van der Vekens, 26-Dec-2017.) (Revised by AV, 9-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       (𝐺𝑈 → (𝐺 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
 
Theoremisushgr 26774 The predicate "is an undirected simple hypergraph." (Contributed by AV, 19-Jan-2020.) (Revised by AV, 9-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       (𝐺𝑈 → (𝐺 ∈ USHGraph ↔ 𝐸:dom 𝐸1-1→(𝒫 𝑉 ∖ {∅})))
 
Theoremuhgrf 26775 The edge function of an undirected hypergraph is a function into the power set of the set of vertices. (Contributed by Alexander van der Vekens, 26-Dec-2017.) (Revised by AV, 9-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       (𝐺 ∈ UHGraph → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))
 
Theoremushgrf 26776 The edge function of an undirected simple hypergraph is a one-to-one function into the power set of the set of vertices. (Contributed by AV, 9-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       (𝐺 ∈ USHGraph → 𝐸:dom 𝐸1-1→(𝒫 𝑉 ∖ {∅}))
 
Theoremuhgrss 26777 An edge is a subset of vertices. (Contributed by Alexander van der Vekens, 26-Dec-2017.) (Revised by AV, 18-Jan-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       ((𝐺 ∈ UHGraph ∧ 𝐹 ∈ dom 𝐸) → (𝐸𝐹) ⊆ 𝑉)
 
Theoremuhgreq12g 26778 If two sets have the same vertices and the same edges, one set is a hypergraph iff the other set is a hypergraph. (Contributed by Alexander van der Vekens, 26-Dec-2017.) (Revised by AV, 18-Jan-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)    &   𝑊 = (Vtx‘𝐻)    &   𝐹 = (iEdg‘𝐻)       (((𝐺𝑋𝐻𝑌) ∧ (𝑉 = 𝑊𝐸 = 𝐹)) → (𝐺 ∈ UHGraph ↔ 𝐻 ∈ UHGraph))
 
Theoremuhgrfun 26779 The edge function of an undirected hypergraph is a function. (Contributed by Alexander van der Vekens, 26-Dec-2017.) (Revised by AV, 15-Dec-2020.)
𝐸 = (iEdg‘𝐺)       (𝐺 ∈ UHGraph → Fun 𝐸)
 
Theoremuhgrn0 26780 An edge is a nonempty subset of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 15-Dec-2020.)
𝐸 = (iEdg‘𝐺)       ((𝐺 ∈ UHGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) → (𝐸𝐹) ≠ ∅)
 
Theoremlpvtx 26781 The endpoints of a loop (which is an edge at index 𝐽) are two (identical) vertices 𝐴. (Contributed by AV, 1-Feb-2021.)
𝐼 = (iEdg‘𝐺)       ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼𝐽) = {𝐴}) → 𝐴 ∈ (Vtx‘𝐺))
 
Theoremushgruhgr 26782 An undirected simple hypergraph is an undirected hypergraph. (Contributed by AV, 19-Jan-2020.) (Revised by AV, 9-Oct-2020.)
(𝐺 ∈ USHGraph → 𝐺 ∈ UHGraph)
 
Theoremisuhgrop 26783 The property of being an undirected hypergraph represented as an ordered pair. The representation as an ordered pair is the usual representation of a graph, see section I.1 of [Bollobas] p. 1. (Contributed by AV, 1-Jan-2020.) (Revised by AV, 9-Oct-2020.)
((𝑉𝑊𝐸𝑋) → (⟨𝑉, 𝐸⟩ ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
 
Theoremuhgr0e 26784 The empty graph, with vertices but no edges, is a hypergraph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 25-Nov-2020.)
(𝜑𝐺𝑊)    &   (𝜑 → (iEdg‘𝐺) = ∅)       (𝜑𝐺 ∈ UHGraph)
 
Theoremuhgr0vb 26785 The null graph, with no vertices, is a hypergraph if and only if the edge function is empty. (Contributed by Alexander van der Vekens, 27-Dec-2017.) (Revised by AV, 9-Oct-2020.)
((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅) → (𝐺 ∈ UHGraph ↔ (iEdg‘𝐺) = ∅))
 
Theoremuhgr0 26786 The null graph represented by an empty set is a hypergraph. (Contributed by AV, 9-Oct-2020.)
∅ ∈ UHGraph
 
Theoremuhgrun 26787 The union 𝑈 of two (undirected) hypergraphs 𝐺 and 𝐻 with the same vertex set 𝑉 is a hypergraph with the vertex 𝑉 and the union (𝐸𝐹) of the (indexed) edges. (Contributed by AV, 11-Oct-2020.) (Revised by AV, 24-Oct-2021.)
(𝜑𝐺 ∈ UHGraph)    &   (𝜑𝐻 ∈ UHGraph)    &   𝐸 = (iEdg‘𝐺)    &   𝐹 = (iEdg‘𝐻)    &   𝑉 = (Vtx‘𝐺)    &   (𝜑 → (Vtx‘𝐻) = 𝑉)    &   (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)    &   (𝜑𝑈𝑊)    &   (𝜑 → (Vtx‘𝑈) = 𝑉)    &   (𝜑 → (iEdg‘𝑈) = (𝐸𝐹))       (𝜑𝑈 ∈ UHGraph)
 
Theoremuhgrunop 26788 The union of two (undirected) hypergraphs (with the same vertex set) represented as ordered pair: If 𝑉, 𝐸 and 𝑉, 𝐹 are hypergraphs, then 𝑉, 𝐸𝐹 is a hypergraph (the vertex set stays the same, but the edges from both graphs are kept, possibly resulting in two edges between two vertices). (Contributed by Alexander van der Vekens, 27-Dec-2017.) (Revised by AV, 11-Oct-2020.) (Revised by AV, 24-Oct-2021.)
(𝜑𝐺 ∈ UHGraph)    &   (𝜑𝐻 ∈ UHGraph)    &   𝐸 = (iEdg‘𝐺)    &   𝐹 = (iEdg‘𝐻)    &   𝑉 = (Vtx‘𝐺)    &   (𝜑 → (Vtx‘𝐻) = 𝑉)    &   (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)       (𝜑 → ⟨𝑉, (𝐸𝐹)⟩ ∈ UHGraph)
 
Theoremushgrun 26789 The union 𝑈 of two (undirected) simple hypergraphs 𝐺 and 𝐻 with the same vertex set 𝑉 is a (not necessarily simple) hypergraph with the vertex 𝑉 and the union (𝐸𝐹) of the (indexed) edges. (Contributed by AV, 29-Nov-2020.) (Revised by AV, 24-Oct-2021.)
(𝜑𝐺 ∈ USHGraph)    &   (𝜑𝐻 ∈ USHGraph)    &   𝐸 = (iEdg‘𝐺)    &   𝐹 = (iEdg‘𝐻)    &   𝑉 = (Vtx‘𝐺)    &   (𝜑 → (Vtx‘𝐻) = 𝑉)    &   (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)    &   (𝜑𝑈𝑊)    &   (𝜑 → (Vtx‘𝑈) = 𝑉)    &   (𝜑 → (iEdg‘𝑈) = (𝐸𝐹))       (𝜑𝑈 ∈ UHGraph)
 
Theoremushgrunop 26790 The union of two (undirected) simple hypergraphs (with the same vertex set) represented as ordered pair: If 𝑉, 𝐸 and 𝑉, 𝐹 are simple hypergraphs, then 𝑉, 𝐸𝐹 is a (not necessarily simple) hypergraph - the vertex set stays the same, but the edges from both graphs are kept, possibly resulting in two edges between two vertices. (Contributed by AV, 29-Nov-2020.) (Revised by AV, 24-Oct-2021.)
(𝜑𝐺 ∈ USHGraph)    &   (𝜑𝐻 ∈ USHGraph)    &   𝐸 = (iEdg‘𝐺)    &   𝐹 = (iEdg‘𝐻)    &   𝑉 = (Vtx‘𝐺)    &   (𝜑 → (Vtx‘𝐻) = 𝑉)    &   (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)       (𝜑 → ⟨𝑉, (𝐸𝐹)⟩ ∈ UHGraph)
 
Theoremuhgrstrrepe 26791 Replacing (or adding) the edges (between elements of the base set) of an extensible structure results in a hypergraph. Instead of requiring (𝜑𝐺 Struct 𝑋), it would be sufficient to require (𝜑 → Fun (𝐺 ∖ {∅})) and (𝜑𝐺 ∈ V). (Contributed by AV, 18-Jan-2020.) (Revised by AV, 7-Jun-2021.) (Revised by AV, 16-Nov-2021.)
𝑉 = (Base‘𝐺)    &   𝐼 = (.ef‘ndx)    &   (𝜑𝐺 Struct 𝑋)    &   (𝜑 → (Base‘ndx) ∈ dom 𝐺)    &   (𝜑𝐸𝑊)    &   (𝜑𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))       (𝜑 → (𝐺 sSet ⟨𝐼, 𝐸⟩) ∈ UHGraph)
 
Theoremincistruhgr 26792* An incidence structure 𝑃, 𝐿, 𝐼 "where 𝑃 is a set whose elements are called points, 𝐿 is a distinct set whose elements are called lines and 𝐼 ⊆ (𝑃 × 𝐿) is the incidence relation" (see Wikipedia "Incidence structure" (24-Oct-2020), https://en.wikipedia.org/wiki/Incidence_structure) implies an undirected hypergraph, if the incidence relation is right-total (to exclude empty edges). The points become the vertices, and the edge function is derived from the incidence relation by mapping each line ("edge") to the set of vertices incident to the line/edge. With 𝑃 = (Base‘𝑆) and by defining two new slots for lines and incidence relations (analogous to LineG and Itv) and enhancing the definition of iEdg accordingly, it would even be possible to express that a corresponding incidence structure is an undirected hypergraph. By choosing the incident relation appropriately, other kinds of undirected graphs (pseudographs, multigraphs, simple graphs, etc.) could be defined. (Contributed by AV, 24-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       ((𝐺𝑊𝐼 ⊆ (𝑃 × 𝐿) ∧ ran 𝐼 = 𝐿) → ((𝑉 = 𝑃𝐸 = (𝑒𝐿 ↦ {𝑣𝑃𝑣𝐼𝑒})) → 𝐺 ∈ UHGraph))
 
16.2.2  Undirected pseudographs and multigraphs
 
Syntaxcupgr 26793 Extend class notation with undirected pseudographs.
class UPGraph
 
Syntaxcumgr 26794 Extend class notation with undirected multigraphs.
class UMGraph
 
Definitiondf-upgr 26795* Define the class of all undirected pseudographs. An (undirected) pseudograph consists of a set 𝑣 (of "vertices") and a function 𝑒 (representing indexed "edges") into subsets of 𝑣 of cardinality one or two, representing the two vertices incident to the edge, or the one vertex if the edge is a loop. This is according to Chartrand, Gary and Zhang, Ping (2012): "A First Course in Graph Theory.", Dover, ISBN 978-0-486-48368-9, section 1.4, p. 26: "In a pseudograph, not only are parallel edges permitted but an edge is also permitted to join a vertex to itself. Such an edge is called a loop." (in contrast to a multigraph, see df-umgr 26796). (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 24-Nov-2020.)
UPGraph = {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}}
 
Definitiondf-umgr 26796* Define the class of all undirected multigraphs. An (undirected) multigraph consists of a set 𝑣 (of "vertices") and a function 𝑒 (representing indexed "edges") into subsets of 𝑣 of cardinality two, representing the two vertices incident to the edge. In contrast to a pseudograph, a multigraph has no loop. This is according to Chartrand, Gary and Zhang, Ping (2012): "A First Course in Graph Theory.", Dover, ISBN 978-0-486-48368-9, section 1.4, p. 26: "A multigraph M consists of a finite nonempty set V of vertices and a set E of edges, where every two vertices of M are joined by a finite number of edges (possibly zero). If two or more edges join the same pair of (distinct) vertices, then these edges are called parallel edges." To provide uniform definitions for all kinds of graphs, 𝑥 ∈ (𝒫 𝑣 ∖ {∅}) is used as restriction of the class abstraction, although 𝑥 ∈ 𝒫 𝑣 would be sufficient (see prprrab 13821 and isumgrs 26809). (Contributed by AV, 24-Nov-2020.)
UMGraph = {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}}
 
Theoremisupgr 26797* The property of being an undirected pseudograph. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 10-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       (𝐺𝑈 → (𝐺 ∈ UPGraph ↔ 𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
 
Theoremwrdupgr 26798* The property of being an undirected pseudograph, expressing the edges as "words". (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 10-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       ((𝐺𝑈𝐸 ∈ Word 𝑋) → (𝐺 ∈ UPGraph ↔ 𝐸 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
 
Theoremupgrf 26799* The edge function of an undirected pseudograph is a function into unordered pairs of vertices. Version of upgrfn 26800 without explicitly specified domain of the edge function. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 10-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       (𝐺 ∈ UPGraph → 𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
 
Theoremupgrfn 26800* The edge function of an undirected pseudograph is a function into unordered pairs of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 10-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴) → 𝐸:𝐴⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44804
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