graph cycle
if 
is a path and 
, then the graph 
is called a cycle.
[cite:;taken from @graph_diestel_2017 chapter 1.3 paths and cycles]
is a path and , then the graph is called a cycle.[cite:;taken from @graph_diestel_2017 chapter 1.3 paths and cycles]
a non-empty sequence 
of darts is a walk if the head of 
is the tail of 
for every 
. to be more specific, it is a 
-to-
walk if is 
or the tail of and is 
or the head of . we define the successor of in 
to be and we define predecessor of to be . we may designate a walk to be a closed walk if the tail of is the head of , in which case we define the successor of to be and the predecesor of to be . we also refer to a closed walk as a tour.
[cite:;taken from @graph_klein_2024 chapter 2.2 walks, paths, and cycles]
of darts is a walk if the head of is the tail of for every . to be more specific, it is a -to- walk if is or the tail of and is or the head of . we define the successor of in to be and we define predecessor of to be . we may designate a walk to be a closed walk if the tail of is the head of , in which case we define the successor of to be and the predecesor of to be . we also refer to a closed walk as a tour.[cite:;taken from @graph_klein_2024 chapter 2.2 walks, paths, and cycles]
a walk is called a path of darts if the darts are distinct, a cycle of darts if in addition it is a closed walk. a pathcycle of darts is called a pathcycle of arcs if each dart is of the form 
. it is called a path/cycle of edges if no edge is represented twice.
[cite:;taken from @graph_klein_2024 chapter 2.2 walks, paths, and cycles]
broken link: xopp-figure:/home/mahmooz/brain/pen/1733222464.8443475.xopp
. it is called a path/cycle of edges if no edge is represented twice.[cite:;taken from @graph_klein_2024 chapter 2.2 walks, paths, and cycles]
let 
be a graph, and let 
be a spanning forest of . for a dart 
of an nontree edge, there is a simple head()-to-tail() path of darts in whose edges belong to . write 
so 
is a simple cycle 
of darts, called the fundamental cycle of with respect to . for an arc 
of 
, we define the fundamental cycle of to be the fundamental cycle of the primary dart .
[cite:;taken from @graph_klein_2024 chapter 3.1.1 nontree edges and fundamental cycles]
be a graph, and let be a spanning forest of . for a dart of an nontree edge, there is a simple head()-to-tail() path of darts in whose edges belong to . write so is a simple cycle of darts, called the fundamental cycle of with respect to . for an arc of , we define the fundamental cycle of to be the fundamental cycle of the primary dart .[cite:;taken from @graph_klein_2024 chapter 3.1.1 nontree edges and fundamental cycles]
broken link: xopp-figure:/home/mahmooz/brain/pen/2024-11-29-Note-23-06.xopp
let 
be a simple cycle of darts in a connected plane graph 
. let 
be an arbitrary face, designated the infinite face. we say the cycle encloses a face 
with respect to if 
where 
.
[cite:;taken from @graph_klein_2024 chapter 4.7 faces, edges, and vertices enclosed by a simple cycle]
be a simple cycle of darts in a connected plane graph . let be an arbitrary face, designated the infinite face. we say the cycle encloses a face with respect to if where .[cite:;taken from @graph_klein_2024 chapter 4.7 faces, edges, and vertices enclosed by a simple cycle]