P. 260 of Theorem List - Metamath Proof Explorer

Type

Label

Description

Statement

Theorem

axcont 25901*

The axiom of continuity. Take two sets of points 𝐴 and 𝐵. If all the points in 𝐴 come before the points of 𝐵 on a line, then there is a point separating the two. Axiom A11 of [Schwabhauser] p. 13. (Contributed by Scott Fenton, 20-Jun-2013.)

⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∃𝑎 ∈ (𝔼‘𝑁)∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑎, 𝑦⟩)) → ∃𝑏 ∈ (𝔼‘𝑁)∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑏 Btwn ⟨𝑥, 𝑦⟩)

15.4.2.3  EE^n fulfills Tarski's Axioms

Syntax

ceeng 25902

Extends class notation with the Tarski geometry structure for 𝔼↑𝑁.

class EEG

Definition

df-eeng 25903*

Define the geometry structure for 𝔼↑𝑁. (Contributed by Thierry Arnoux, 24-Aug-2017.)

⊢ EEG = (𝑛 ∈ ℕ ↦ ({⟨(Base‘ndx), (𝔼‘𝑛)⟩, ⟨(dist‘ndx), (𝑥 ∈ (𝔼‘𝑛), 𝑦 ∈ (𝔼‘𝑛) ↦ Σ𝑖 ∈ (1...𝑛)(((𝑥‘𝑖) − (𝑦‘𝑖))↑2))⟩} ∪ {⟨(Itv‘ndx), (𝑥 ∈ (𝔼‘𝑛), 𝑦 ∈ (𝔼‘𝑛) ↦ {𝑧 ∈ (𝔼‘𝑛) ∣ 𝑧 Btwn ⟨𝑥, 𝑦⟩})⟩, ⟨(LineG‘ndx), (𝑥 ∈ (𝔼‘𝑛), 𝑦 ∈ ((𝔼‘𝑛) ∖ {𝑥}) ↦ {𝑧 ∈ (𝔼‘𝑛) ∣ (𝑧 Btwn ⟨𝑥, 𝑦⟩ ∨ 𝑥 Btwn ⟨𝑧, 𝑦⟩ ∨ 𝑦 Btwn ⟨𝑥, 𝑧⟩)})⟩}))

Theorem

eengv 25904*

The value of the Euclidean geometry for dimension 𝑁. (Contributed by Thierry Arnoux, 15-Mar-2019.)

⊢ (𝑁 ∈ ℕ → (EEG‘𝑁) = ({⟨(Base‘ndx), (𝔼‘𝑁)⟩, ⟨(dist‘ndx), (𝑥 ∈ (𝔼‘𝑁), 𝑦 ∈ (𝔼‘𝑁) ↦ Σ𝑖 ∈ (1...𝑁)(((𝑥‘𝑖) − (𝑦‘𝑖))↑2))⟩} ∪ {⟨(Itv‘ndx), (𝑥 ∈ (𝔼‘𝑁), 𝑦 ∈ (𝔼‘𝑁) ↦ {𝑧 ∈ (𝔼‘𝑁) ∣ 𝑧 Btwn ⟨𝑥, 𝑦⟩})⟩, ⟨(LineG‘ndx), (𝑥 ∈ (𝔼‘𝑁), 𝑦 ∈ ((𝔼‘𝑁) ∖ {𝑥}) ↦ {𝑧 ∈ (𝔼‘𝑁) ∣ (𝑧 Btwn ⟨𝑥, 𝑦⟩ ∨ 𝑥 Btwn ⟨𝑧, 𝑦⟩ ∨ 𝑦 Btwn ⟨𝑥, 𝑧⟩)})⟩}))

Theorem

eengstr 25905

The Euclidean geometry as a structure. (Contributed by Thierry Arnoux, 15-Mar-2019.)

⊢ (𝑁 ∈ ℕ → (EEG‘𝑁) Struct ⟨1, ;17⟩)

Theorem

eengbas 25906

The Base of the Euclidean geometry. (Contributed by Thierry Arnoux, 15-Mar-2019.)

⊢ (𝑁 ∈ ℕ → (𝔼‘𝑁) = (Base‘(EEG‘𝑁)))

Theorem

ebtwntg 25907

The betweenness relation used in the Tarski structure for the Euclidean geometry is the same as Btwn. (Contributed by Thierry Arnoux, 15-Mar-2019.)

⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ 𝑃 = (Base‘(EEG‘𝑁))    &   ⊢ 𝐼 = (Itv‘(EEG‘𝑁))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑃)    ⇒   ⊢ (𝜑 → (𝑍 Btwn ⟨𝑋, 𝑌⟩ ↔ 𝑍 ∈ (𝑋𝐼𝑌)))

Theorem

ecgrtg 25908

The congruence relation used in the Tarski structure for the Euclidean geometry is the same as Cgr. (Contributed by Thierry Arnoux, 15-Mar-2019.)

⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ 𝑃 = (Base‘(EEG‘𝑁))    &   ⊢ − = (dist‘(EEG‘𝑁))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑃)    ⇒   ⊢ (𝜑 → (⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝐷⟩ ↔ (𝐴 − 𝐵) = (𝐶 − 𝐷)))

Theorem

elntg 25909*

The line definition in the Tarski structure for the Euclidean geometry. (Contributed by Thierry Arnoux, 7-Apr-2019.)

⊢ 𝑃 = (Base‘(EEG‘𝑁))    &   ⊢ 𝐼 = (Itv‘(EEG‘𝑁))    ⇒   ⊢ (𝑁 ∈ ℕ → (LineG‘(EEG‘𝑁)) = (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}))

Theorem

eengtrkg 25910

The geometry structure for 𝔼↑𝑁 is a Tarski geometry. (Contributed by Thierry Arnoux, 15-Mar-2019.)

⊢ (𝑁 ∈ ℕ → (EEG‘𝑁) ∈ TarskiG)

Theorem

eengtrkge 25911

The geometry structure for 𝔼↑𝑁 is a Euclidean geometry. (Contributed by Thierry Arnoux, 15-Mar-2019.)

⊢ (𝑁 ∈ ℕ → (EEG‘𝑁) ∈ TarskiG_E)

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:

Term	Reference	Definition	Remarks
Vertex	df-vtx 25921	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 25929), or the base set (Base‘𝐺) of the graph 𝐺 if it is represented as extensible structure (see basvtxval 25946).
Edge	df-edg 25985	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 25922	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 25932), or the component (.ef‘𝐺) of the graph 𝐺 if it is represented as extensible structure (see edgfiedgval 25947). The set of edges of a graph 𝐺 is the range of its edge function: (Edg‘𝐺) = ran (iEdg‘𝐺), see edgval 25986. 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:

Term	Reference	Definition	Remarks
Undirected hypergraph	df-uhgr 25998	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 (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 (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 25999	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 (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 26022	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 4424). This means a pseudograph may contain loops. This defintion 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 26023	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 26090	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 26091	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 26109.
Finite graph	df-fusgr 26254	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 26267. The number of edges is limited by (𝑛C2) (or "𝑛 choose 2") with 𝑛 = (#‘𝑉), see fusgrmaxsize 26416. 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:

Term	Reference	Definition	Remarks
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: 𝑣 ∈ 𝑒. 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 26270 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 26418	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 26485	A vertex is called isolated if it is not an endvertex of any edge, thus having degree 0.
Universal vertex	df-uvtx 26332	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:

Term	Reference	Definition	Remarks
Complete graph	df-cplgr 26362	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 26406.
Empty graph	uhgr0e 26011	A graph is called empty if it has no edges.
Null graph	uhgr0 26013 and uhgr0vb 26012	A graph is called a null graph if it has no vertices (and therefore also no edges).
Trivial graph	usgr1v 26193	A graph is called the "trivial graph" if it has only one vertex and no edges.
Connected graph	df-conngr 27165 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 25929), or the base set (Base‘𝐺) of the graph 𝐺 if it is represented as extensible structure (see basvtxval 25946), 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 25932), or the component (.ef‘𝐺) of the graph 𝐺 if it is represented as extensible structure (see edgfiedgval 25947). 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 25986.

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 25982 and iedgvalprc 25983.

Up to the end of this section, the edges need not be related to the vertices. Once undirected hypergraphs are defined (see df-uhgr 25998), 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

Syntax

cedgf 25912

Extend class notation with an edge function.

class .ef

Definition

df-edgf 25913

Define the edge function (indexed edges) of a graph. (Contributed by AV, 18-Jan-2020.)

⊢ .ef = Slot ;18

Theorem

edgfid 25914

Utility theorem: index-independent form of df-edgf 25913. (Contributed by AV, 16-Nov-2021.)

⊢ .ef = Slot (.ef‘ndx)

Theorem

edgfndxnn 25915

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

Theorem

edgfndxid 25916

The value of the edge function extractor is the value of the corresponding slot of the structure. (Contributed by AV, 21-Sep-2020.)

⊢ (𝐺 ∈ 𝑉 → (.ef‘𝐺) = (𝐺‘(.ef‘ndx)))

Theorem

baseltedgf 25917

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)

Theorem

slotsbaseefdif 25918

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 15906.

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 25921 and df-iedg 25922), 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) = (1^st ‘𝐺) (see opvtxval 25928) and the set of (indexed) edges is (iEdg‘𝐺) = (2^nd ‘𝐺) (see opiedgval 25931), or if 𝐺 is given as ordered pair 𝐺 = ⟨𝑉, 𝐸⟩, the set of vertices is (Vtx‘𝐺) = 𝑉 (see opvtxfv 25929) and the set of (indexed) edges is (iEdg‘𝐺) = 𝐸 (see opiedgfv 25932).

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 25950) and the set of (indexed) edges is (iEdg‘𝐺) = (.ef‘𝐺) (see funiedgval 25951), 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 25964) and the set of (indexed) edges is (iEdg‘𝐺) = 𝐸 (see struct2griedg 25965).

These two representations are convertible, see graop 25966 and grastruct 25967: 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 25968 and gropeld 25970 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 25969 and grstructeld 25971 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 25974 and snstriedgval 25975. Analogously, the empty set ∅ can be used to represent the null graph, see vtxval0 25976 and iedgval0 25977, 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 25982 and iedgvalprc 25983.

Other classes should not be used to represent graphs, because there could be a degenerated behavior of the vertex set and (indexed) edge functions, see vtxvalsnop 25978 resp. iedgvalsnop 25979, and vtxval3sn 25980 resp. iedgval3sn 25981.

16.1.2.1  Definitions and basic properties

Syntax

cvtx 25919

Extend class notation with the vertices of "graphs".

class Vtx

Syntax

ciedg 25920

Extend class notation with the indexed edges of "graphs".

class iEdg

Definition

df-vtx 25921

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), (1^st ‘𝑔), (Base‘𝑔)))

Definition

df-iedg 25922

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), (2^nd ‘𝑔), (.ef‘𝑔)))

Theorem

vtxval 25923

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), (1^st ‘𝐺), (Base‘𝐺))

Theorem

iedgval 25924

The set of indexed edges of a graph. (Contributed by AV, 21-Sep-2020.)

⊢ (iEdg‘𝐺) = if(𝐺 ∈ (V × V), (2^nd ‘𝐺), (.ef‘𝐺))

Theorem

vtxvalOLD 25925

Obsolete version of vtxval 25923 as of 11-Nov-2021. (Contributed by AV, 9-Jan-2020.) (Revised by AV, 21-Sep-2020.) (New usage is discouraged.) (Proof modification is discouraged.)

⊢ (𝐺 ∈ 𝑉 → (Vtx‘𝐺) = if(𝐺 ∈ (V × V), (1^st ‘𝐺), (Base‘𝐺)))

Theorem

iedgvalOLD 25926

Obsolete version of iedgval 25924 as of 11-Nov-2021. (Contributed by AV, 21-Sep-2020.) (New usage is discouraged.) (Proof modification is discouraged.)

⊢ (𝐺 ∈ 𝑉 → (iEdg‘𝐺) = if(𝐺 ∈ (V × V), (2^nd ‘𝐺), (.ef‘𝐺)))

Theorem

1vgrex 25927

A graph with at least one vertex is a set. (Contributed by AV, 2-Mar-2021.)

⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝑁 ∈ 𝑉 → 𝐺 ∈ V)

16.1.2.2  The vertices and edges of a graph represented as ordered pair

Theorem

opvtxval 25928

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) = (1^st ‘𝐺))

Theorem

opvtxfv 25929

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.)

⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉)

Theorem

opvtxov 25930

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.)

⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑉Vtx𝐸) = 𝑉)

Theorem

opiedgval 25931

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) = (2^nd ‘𝐺))

Theorem

opiedgfv 25932

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.)

⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)

Theorem

opiedgov 25933

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.)

⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑉iEdg𝐸) = 𝐸)

Theorem

opvtxfvi 25934

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

Theorem

opiedgfvi 25935

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

Theorem

funvtxdmge2val 25936

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))

Theorem

funiedgdmge2val 25937

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))

Theorem

funvtxdm2val 25938

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 ∖ {∅}) ∧ 𝐴 ≠ 𝐵 ∧ {𝐴, 𝐵} ⊆ dom 𝐺) → (Vtx‘𝐺) = (Base‘𝐺))

Theorem

funiedgdm2val 25939

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 ∖ {∅}) ∧ 𝐴 ≠ 𝐵 ∧ {𝐴, 𝐵} ⊆ dom 𝐺) → (iEdg‘𝐺) = (.ef‘𝐺))

Theorem

funvtxdm2valOLD 25940

Obsolete version of funvtxdm2val 25938 as of 11-Nov-2021. (Contributed by AV, 22-Sep-2020.) (Revised by AV, 7-Jun-2021.) (New usage is discouraged.) (Proof modification is discouraged.)

⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (((𝐺 ∈ 𝑉 ∧ Fun (𝐺 ∖ {∅})) ∧ 𝐴 ≠ 𝐵 ∧ {𝐴, 𝐵} ⊆ dom 𝐺) → (Vtx‘𝐺) = (Base‘𝐺))

Theorem

funiedgdm2valOLD 25941

Obsolete version of funiedgdm2val 25939 as of 11-Nov-2021. (Contributed by AV, 22-Sep-2020.) (Revised by AV, 7-Jun-2021.) (New usage is discouraged.) (Proof modification is discouraged.)

⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (((𝐺 ∈ 𝑉 ∧ Fun (𝐺 ∖ {∅})) ∧ 𝐴 ≠ 𝐵 ∧ {𝐴, 𝐵} ⊆ dom 𝐺) → (iEdg‘𝐺) = (.ef‘𝐺))

Theorem

funvtxval0 25942

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))

Theorem

funvtxval0OLD 25943

Obsolete version of funvtxval0 25942 as of 11-Nov-2021. (Contributed by AV, 22-Sep-2020.) (Revised by AV, 7-Jun-2021.) (New usage is discouraged.) (Proof modification is discouraged.)

⊢ 𝑆 ∈ V    ⇒   ⊢ (((𝐺 ∈ 𝑉 ∧ Fun (𝐺 ∖ {∅})) ∧ 𝑆 ≠ (Base‘ndx) ∧ {(Base‘ndx), 𝑆} ⊆ dom 𝐺) → (Vtx‘𝐺) = (Base‘𝐺))

Theorem

funvtxdmge2valOLD 25944

Obsolete version of funvtxdmge2val 25936 as of 11-Nov-2021. (Contributed by AV, 12-Oct-2020.) (Revised by AV, 7-Jun-2021.) (New usage is discouraged.) (Proof modification is discouraged.)

⊢ ((𝐺 ∈ 𝑉 ∧ Fun (𝐺 ∖ {∅}) ∧ 2 ≤ (#‘dom 𝐺)) → (Vtx‘𝐺) = (Base‘𝐺))

Theorem

funiedgdmge2valOLD 25945

Obsolete version of funiedgdmge2val 25937 as of 11-Nov-2021. (Contributed by AV, 12-Oct-2020.) (Revised by AV, 7-Jun-2021.) (New usage is discouraged.) (Proof modification is discouraged.)

⊢ ((𝐺 ∈ 𝑉 ∧ Fun (𝐺 ∖ {∅}) ∧ 2 ≤ (#‘dom 𝐺)) → (iEdg‘𝐺) = (.ef‘𝐺))

Theorem

basvtxval 25946

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.)

⊢ (𝜑 → 𝐺 Struct 𝑋)    &   ⊢ (𝜑 → 2 ≤ (#‘dom 𝐺))    &   ⊢ (𝜑 → 𝑉 ∈ 𝑌)    &   ⊢ (𝜑 → ⟨(Base‘ndx), 𝑉⟩ ∈ 𝐺)    ⇒   ⊢ (𝜑 → (Vtx‘𝐺) = 𝑉)

Theorem

edgfiedgval 25947

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.)

⊢ (𝜑 → 𝐺 Struct 𝑋)    &   ⊢ (𝜑 → 2 ≤ (#‘dom 𝐺))    &   ⊢ (𝜑 → 𝐸 ∈ 𝑌)    &   ⊢ (𝜑 → ⟨(.ef‘ndx), 𝐸⟩ ∈ 𝐺)    ⇒   ⊢ (𝜑 → (iEdg‘𝐺) = 𝐸)

Theorem

basvtxvalOLD 25948

Obsolete version of basvtxval 25946 as of 12-Nov-2021. (Contributed by AV, 14-Oct-2020.) (New usage is discouraged.) (Proof modification is discouraged.)

⊢ (𝜑 → 𝐺 ∈ 𝑋)    &   ⊢ (𝜑 → Fun 𝐺)    &   ⊢ (𝜑 → 2 ≤ (#‘dom 𝐺))    &   ⊢ (𝜑 → 𝑉 ∈ 𝑌)    &   ⊢ (𝜑 → ⟨(Base‘ndx), 𝑉⟩ ∈ 𝐺)    ⇒   ⊢ (𝜑 → (Vtx‘𝐺) = 𝑉)

Theorem

edgfiedgvalOLD 25949

Obsolete version of edgfiedgval 25947 as of 12-Nov-2021. (Contributed by AV, 14-Oct-2020.) (New usage is discouraged.) (Proof modification is discouraged.)

⊢ (𝜑 → 𝐺 ∈ 𝑋)    &   ⊢ (𝜑 → Fun 𝐺)    &   ⊢ (𝜑 → 2 ≤ (#‘dom 𝐺))    &   ⊢ (𝜑 → 𝐸 ∈ 𝑌)    &   ⊢ (𝜑 → ⟨(.ef‘ndx), 𝐸⟩ ∈ 𝐺)    ⇒   ⊢ (𝜑 → (iEdg‘𝐺) = 𝐸)

Theorem

funvtxval 25950

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))

Theorem

funiedgval 25951

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))

Theorem

funvtxvalOLD 25952

Obsolete version of funvtxval 25950 as of 12-Nov-2021. (Contributed by AV, 22-Sep-2020.) (Revised by AV, 7-Jun-2021.) (New usage is discouraged.) (Proof modification is discouraged.)

⊢ ((𝐺 ∈ 𝑉 ∧ Fun (𝐺 ∖ {∅}) ∧ {(Base‘ndx), (.ef‘ndx)} ⊆ dom 𝐺) → (Vtx‘𝐺) = (Base‘𝐺))

Theorem

funiedgvalOLD 25953

Obsolete version of funiedgval 25951 as of 12-Nov-2021. (Contributed by AV, 21-Sep-2020.) (Revised by AV, 7-Jun-2021.) (New usage is discouraged.) (Proof modification is discouraged.)

⊢ ((𝐺 ∈ 𝑉 ∧ Fun (𝐺 ∖ {∅}) ∧ {(Base‘ndx), (.ef‘ndx)} ⊆ dom 𝐺) → (iEdg‘𝐺) = (.ef‘𝐺))

Theorem

structvtxvallem 25954

Lemma for structvtxval 25955 and structiedg0val 25956. (Contributed by AV, 23-Sep-2020.) (Revised by AV, 12-Nov-2021.)

⊢ 𝑆 ∈ ℕ    &   ⊢ (Base‘ndx) < 𝑆    &   ⊢ 𝐺 = {⟨(Base‘ndx), 𝑉⟩, ⟨𝑆, 𝐸⟩}    ⇒   ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → 2 ≤ (#‘dom 𝐺))

Theorem

structvtxval 25955

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⟩}    ⇒   ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (Vtx‘𝐺) = 𝑉)

Theorem

structiedg0val 25956

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⟩}    ⇒   ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑆 ≠ (.ef‘ndx)) → (iEdg‘𝐺) = ∅)

Theorem

structgrssvtxlem 25957

Lemma for structgrssvtx 25958 and structgrssiedg 25959. (Contributed by AV, 14-Oct-2020.) (Proof shortened by AV, 12-Nov-2021.)

⊢ (𝜑 → 𝐺 Struct 𝑋)    &   ⊢ (𝜑 → 𝑉 ∈ 𝑌)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑍)    &   ⊢ (𝜑 → {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩} ⊆ 𝐺)    ⇒   ⊢ (𝜑 → 2 ≤ (#‘dom 𝐺))

Theorem

structgrssvtx 25958

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.)

⊢ (𝜑 → 𝐺 Struct 𝑋)    &   ⊢ (𝜑 → 𝑉 ∈ 𝑌)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑍)    &   ⊢ (𝜑 → {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩} ⊆ 𝐺)    ⇒   ⊢ (𝜑 → (Vtx‘𝐺) = 𝑉)

Theorem

structgrssiedg 25959

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.)

⊢ (𝜑 → 𝐺 Struct 𝑋)    &   ⊢ (𝜑 → 𝑉 ∈ 𝑌)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑍)    &   ⊢ (𝜑 → {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩} ⊆ 𝐺)    ⇒   ⊢ (𝜑 → (iEdg‘𝐺) = 𝐸)

Theorem

structgrssvtxlemOLD 25960

Obsolete version of structgrssvtxlem 25957 as of 14-Nov-2021. (Contributed by AV, 14-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜑 → 𝐺 ∈ 𝑋)    &   ⊢ (𝜑 → Fun 𝐺)    &   ⊢ (𝜑 → 𝑉 ∈ 𝑌)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑍)    &   ⊢ (𝜑 → {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩} ⊆ 𝐺)    ⇒   ⊢ (𝜑 → 2 ≤ (#‘dom 𝐺))

Theorem

structgrssvtxOLD 25961

Obsolete version of structgrssvtx 25958 as of 14-Nov-2021. (Contributed by AV, 14-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜑 → 𝐺 ∈ 𝑋)    &   ⊢ (𝜑 → Fun 𝐺)    &   ⊢ (𝜑 → 𝑉 ∈ 𝑌)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑍)    &   ⊢ (𝜑 → {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩} ⊆ 𝐺)    ⇒   ⊢ (𝜑 → (Vtx‘𝐺) = 𝑉)

Theorem

structgrssiedgOLD 25962

Obsolete version of structgrssiedg 25959 as of 14-Nov-2021. (Contributed by AV, 14-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜑 → 𝐺 ∈ 𝑋)    &   ⊢ (𝜑 → Fun 𝐺)    &   ⊢ (𝜑 → 𝑉 ∈ 𝑌)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑍)    &   ⊢ (𝜑 → {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩} ⊆ 𝐺)    ⇒   ⊢ (𝜑 → (iEdg‘𝐺) = 𝐸)

Theorem

struct2grstr 25963

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)⟩

Theorem

struct2grvtx 25964

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⟩}    ⇒   ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (Vtx‘𝐺) = 𝑉)

Theorem

struct2griedg 25965

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⟩}    ⇒   ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (iEdg‘𝐺) = 𝐸)

Theorem

graop 25966

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))

Theorem

grastruct 25967

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))

Theorem

gropd 25968*

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))    &   ⊢ (𝜑 → 𝑉 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑊)    ⇒   ⊢ (𝜑 → [⟨𝑉, 𝐸⟩ / 𝑔]𝜓)

Theorem

grstructd 25969*

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))    &   ⊢ (𝜑 → 𝑉 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑋)    &   ⊢ (𝜑 → Fun (𝑆 ∖ {∅}))    &   ⊢ (𝜑 → 2 ≤ (#‘dom 𝑆))    &   ⊢ (𝜑 → (Base‘𝑆) = 𝑉)    &   ⊢ (𝜑 → (.ef‘𝑆) = 𝐸)    ⇒   ⊢ (𝜑 → [𝑆 / 𝑔]𝜓)

Theorem

gropeld 25970*

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) → 𝑔 ∈ 𝐶))    &   ⊢ (𝜑 → 𝑉 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑊)    ⇒   ⊢ (𝜑 → ⟨𝑉, 𝐸⟩ ∈ 𝐶)

Theorem

grstructeld 25971*

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) → 𝑔 ∈ 𝐶))    &   ⊢ (𝜑 → 𝑉 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑋)    &   ⊢ (𝜑 → Fun (𝑆 ∖ {∅}))    &   ⊢ (𝜑 → 2 ≤ (#‘dom 𝑆))    &   ⊢ (𝜑 → (Base‘𝑆) = 𝑉)    &   ⊢ (𝜑 → (.ef‘𝑆) = 𝐸)    ⇒   ⊢ (𝜑 → 𝑆 ∈ 𝐶)

Theorem

setsvtx 25972

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)    &   ⊢ (𝜑 → 𝐺 Struct 𝑋)    &   ⊢ (𝜑 → (Base‘ndx) ∈ dom 𝐺)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (Vtx‘(𝐺 sSet ⟨𝐼, 𝐸⟩)) = (Base‘𝐺))

Theorem

setsiedg 25973

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)    &   ⊢ (𝜑 → 𝐺 Struct 𝑋)    &   ⊢ (𝜑 → (Base‘ndx) ∈ dom 𝐺)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (iEdg‘(𝐺 sSet ⟨𝐼, 𝐸⟩)) = 𝐸)

16.1.2.4  Representations of graphs without edges

Theorem

snstrvtxval 25974

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 25978 for the (degenerated) case where 𝑉 = (Base‘ndx). (Contributed by AV, 23-Sep-2020.)

⊢ 𝑉 ∈ V    &   ⊢ 𝐺 = {⟨(Base‘ndx), 𝑉⟩}    ⇒   ⊢ (𝑉 ≠ (Base‘ndx) → (Vtx‘𝐺) = 𝑉)

Theorem

snstriedgval 25975

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 25979 for the (degenerated) 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

Theorem

vtxval0 25976

Degenerated case 1 for vertices: The set of vertices of the empty set is the empty set. (Contributed by AV, 24-Sep-2020.)

⊢ (Vtx‘∅) = ∅

Theorem

iedgval0 25977

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‘∅) = ∅

Theorem

vtxvalsnop 25978

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, 12-Nov-2021.)

⊢ 𝐵 ∈ V    &   ⊢ 𝐺 = {⟨𝐵, 𝐵⟩}    ⇒   ⊢ (Vtx‘𝐺) = {𝐵}

Theorem

iedgvalsnop 25979

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, 12-Nov-2021.)

⊢ 𝐵 ∈ V    &   ⊢ 𝐺 = {⟨𝐵, 𝐵⟩}    ⇒   ⊢ (iEdg‘𝐺) = {𝐵}

Theorem

vtxval3sn 25980

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.)

⊢ 𝐴 ∈ V    ⇒   ⊢ (Vtx‘{{{𝐴}}}) = {𝐴}

Theorem

iedgval3sn 25981

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.)

⊢ 𝐴 ∈ V    ⇒   ⊢ (iEdg‘{{{𝐴}}}) = {𝐴}

Theorem

vtxvalprc 25982

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) = ∅)

Theorem

iedgvalprc 25983

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

Syntax

cedg 25984

Extend class notation with the set of edges (of an undirected simple (hyper-/pseudo-)graph).

class Edg

Definition

df-edg 25985

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 26069). 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))

Theorem

edgval 25986

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)

Theorem

edgvalOLD 25987

Obsolete version of edgval 25986 as of 8-Dec-2021. (Contributed by AV, 1-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)

⊢ (𝐺 ∈ 𝑉 → (Edg‘𝐺) = ran (iEdg‘𝐺))

Theorem

iedgedg 25988

An indexed edge is an edge. (Contributed by AV, 19-Dec-2021.)

⊢ 𝐸 = (iEdg‘𝐺)    ⇒   ⊢ ((Fun 𝐸 ∧ 𝐼 ∈ dom 𝐸) → (𝐸‘𝐼) ∈ (Edg‘𝐺))

Theorem

edgopval 25989

The edges of a graph represented as ordered pair. (Contributed by AV, 1-Jan-2020.) (Revised by AV, 13-Oct-2020.)

⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (Edg‘⟨𝑉, 𝐸⟩) = ran 𝐸)

Theorem

edgov 25990

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 25989. 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.)

⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (𝑉Edg𝐸) = ran 𝐸)

Theorem

edgstruct 25991

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⟩}    ⇒   ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (Edg‘𝐺) = ran 𝐸)

Theorem

edgiedgb 25992*

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 = (𝐼‘𝑥)))

Theorem

edgiedgbOLD 25993*

Obsolete version of edgiedgb 25992 as of 8-Dec-2021. (Contributed by AV, 17-Oct-2020.) (New usage is discouraged.) (Proof modification is discouraged.)

⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐼) → (𝐸 ∈ (Edg‘𝐺) ↔ ∃𝑥 ∈ dom 𝐼 𝐸 = (𝐼‘𝑥)))

Theorem

edg0iedg0 25994

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 = ∅))

Theorem

edg0iedg0OLD 25995

Obsolete version of edg0iedg0 25994 as of 8-Dec-2021. (Contributed by AV, 15-Dec-2020.) (New usage is discouraged.) (Proof modification is discouraged.)

⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ 𝑊 ∧ 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 26009)

* Undirected Pseudograph: UPGraph => UPGraph ⊆ UHGraph (upgruhgr 26042)

* 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 26116) and USPGraph ⊆ USHGraph (uspgrushgr 26115), see also uspgrupgrushgr 26117

* Undirected Muligraph: UMGraph => UMGraph ⊆ UPGraph (umgrupgr 26043) and UMGraph ⊆ ULFHGraph (umgrislfupgr 26063)

* Undirected simple Graph: USGraph => USGraph ⊆ USPGraph (usgruspgr 26118) and USGraph ⊆ UMGraph (usgrumgr 26119) and USGraph ⊆ ULFSHGraph (usgrislfuspgr 26124) see also usgrumgruspgr 26120

16.2.1  Undirected hypergraphs

Syntax

cuhgr 25996

Extend class notation with undirected hypergraphs.

class UHGraph

Syntax

cushgr 25997

Extend class notation with undirected simple hypergraphs.

class USHGraph

Definition

df-uhgr 25998*

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 ∖ {∅})}

Definition

df-ushgr 25999*

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 non-empty 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 𝑒–1-1→(𝒫 𝑣 ∖ {∅})}

Theorem

isuhgr 26000

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)    ⇒   ⊢ (𝐺 ∈ 𝑈 → (𝐺 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))