# Revision history [back]

### all_simple_paths, but without edge crossings?

I am trying to use sage to list all simple paths without self-intersections between two fixed vertices of certain (embeddings of) graphs. I can use all_simple_paths to output all simple paths from one vertex to the other by creating a dictionary d={} (specifying vertices, and the outgoing edges), then forming a digraph, then using all_simple_paths. But I'm not sure how to restrict the output list so that only simple paths without self-crossings are given.

My idea is to somehow tell sage "if you cross edge j while forming a path, do not use list of edges that cross j while creating the remainder of the path". I am new to sage (and have no programming background), so I'm not sure how to do this, or if this would even work. Any help is really appreciated.

### all_simple_paths, but without edge crossings?

I am trying to use sage to list all simple paths without self-intersections between two fixed vertices of certain (embeddings of) graphs. graphs, but with these two additional conditions:

1. only paths of maximal length are output, and

2. of the maximal length paths from 1., only the paths without self-intersections are kept.

By self-intersection, I mean one edge of the path crossing another (which depends on the embedding of the underlying graph).

I can use all_simple_paths to output all simple paths from one vertex to the other by creating a dictionary d={} (specifying vertices, and the outgoing edges), then forming a digraph, then using all_simple_paths. But I'm not sure how to restrict the output list so that only simple paths both of maximal length and without self-crossings are given.

My Rough idea for 1: I think you can set the maximum length of the paths all_simple_paths outputs, but I didn't see any way to set the minimum length. The ability to set the minimum length of a path would be sufficient for my application.

Rough idea for 2: For excluding self-crossing paths, my idea is to somehow tell sage "if you cross edge j while forming a path, do not use list of edges that cross j while creating the remainder of the path". Since the edge crossings in the underlying graph depends on how we draw it, I would think we have to manually tell sage which edges not to cross and when.

I am new to sage (and have no programming background), so I'm not sure how to do this, or if this would even work. Any help is really appreciated.

Example:

d = {1:[2,4,5], 2:[3,4,5,6], 3:[6], 4:[2,5], 5:[3,4,6]}
D = DiGraph(d)
D.all_simple_paths(starting_vertices=[1], ending_vertices=[6])


I'm drawing this graph as two squares, glued together along one edge, with an "X" inside each square (which is what sage output when I tried G=Graph(d), then show(G), and this is the embedding I had in mind).

This gives the following output:

[[1, 2, 6],
[1, 5, 6],
[1, 2, 3, 6],
[1, 2, 5, 6],
[1, 4, 2, 6],
[1, 4, 5, 6],
[1, 5, 3, 6],
[1, 2, 4, 5, 6],
[1, 2, 5, 3, 6],
[1, 4, 2, 3, 6],
[1, 4, 2, 5, 6],
[1, 4, 5, 3, 6],
[1, 5, 4, 2, 6],
[1, 2, 4, 5, 3, 6],
[1, 4, 2, 5, 3, 6],
[1, 5, 4, 2, 3, 6]]


First, I would like to filter this output so that only the maximal length paths are given:

[[1, 2, 4, 5, 3, 6],
[1, 4, 2, 5, 3, 6],
[1, 5, 4, 2, 3, 6]]


Finally, note that the last maximal path [1, 5, 4, 2, 3, 6] has a self-intersection (according to how we are drawing the underlying graph): the first edge 1-->5 will be crossed by the edge 4-->2 later in the path. As the first two maximal paths listed above do not have any self-intersections, the final output should be:

[[1, 2, 4, 5, 3, 6],
[1, 4, 2, 5, 3, 6]]


(edited for clarity and specificity, and added the example at the end)

### all_simple_paths, but without edge crossings?

I am trying to use sage to list all simple paths between two fixed vertices of certain (embeddings of) graphs, but with these two additional conditions:

1. only paths of maximal length are output, and

2. of the maximal length paths from 1., only the paths without self-intersections are kept.

By self-intersection, I mean one edge of the path crossing another (which depends on the embedding of the underlying graph).

I can use all_simple_paths to output all simple paths from one vertex to the other by creating a dictionary d={} (specifying vertices, and the outgoing edges), then forming a digraph, then using all_simple_paths. But I'm not sure how to restrict the output list so that only simple paths both of maximal length and without self-crossings are given.

Rough idea for 1: I think you can set the maximum length of the paths all_simple_paths outputs, but I didn't see any way to set the minimum length. The ability to set the minimum length of a path would be sufficient for my application.

Rough idea for 2: For excluding self-crossing paths, my idea is to somehow tell sage "if you cross edge j while forming a path, do not use list of edges that cross j while creating the remainder of the path". Since the edge crossings in the underlying graph depends on how we draw it, I would think we have to manually tell sage which edges not to cross and when.

I am new to sage (and have no programming background), so I'm not sure how to do this, or if this would even work. Any help is really appreciated.

Example:

d = {1:[2,4,5], 2:[3,4,5,6], 3:[6], 4:[2,5], 5:[3,4,6]}
D = DiGraph(d)
D.all_simple_paths(starting_vertices=[1], ending_vertices=[6])


I'm drawing this graph as two squares, glued together along one edge, with an "X" inside each square (which is what sage output when I tried G=Graph(d), then show(G), and this is the embedding I had in mind).

This gives the following output:

[[1, 2, 6],
[1, 5, 6],
[1, 2, 3, 6],
[1, 2, 5, 6],
[1, 4, 2, 6],
[1, 4, 5, 6],
[1, 5, 3, 6],
[1, 2, 4, 5, 6],
[1, 2, 5, 3, 6],
[1, 4, 2, 3, 6],
[1, 4, 2, 5, 6],
[1, 4, 5, 3, 6],
[1, 5, 4, 2, 6],
[1, 2, 4, 5, 3, 6],
[1, 4, 2, 5, 3, 6],
[1, 5, 4, 2, 3, 6]]


First, I would like to filter this output so that only the maximal length paths are given:

[[1, 2, 4, 5, 3, 6],
[1, 4, 2, 5, 3, 6],
[1, 5, 4, 2, 3, 6]]


Finally, note that the last maximal path [1, 5, 4, 2, 3, 6] has a self-intersection (according to how we are drawing the underlying graph): the first edge 1-->5 will be crossed by the edge 4-->2 later in the path. As the first two maximal paths listed above do not have any self-intersections, the final output should be:

[[1, 2, 4, 5, 3, 6],
[1, 4, 2, 5, 3, 6]]


(edited for clarity and specificity, and added the example at the end)

### Maximal paths in all_simple_paths, but and without edge crossings?

I am trying to use sage to list all simple paths between two fixed vertices of certain (embeddings of) graphs, but with these two additional conditions:

1. only paths of maximal length are output, and

2. of the maximal length paths from 1., only the paths without self-intersections are kept.

By self-intersection, I mean one edge of the path crossing another (which depends on the embedding of the underlying graph).

I can use all_simple_paths to output all simple paths from one vertex to the other by creating a dictionary d={} (specifying vertices, and the outgoing edges), then forming a digraph, then using all_simple_paths. But I'm not sure how to restrict the output list so that only simple paths both of maximal length and without self-crossings are given.

Rough idea for 1: I think you can set the maximum length of the paths all_simple_paths outputs, but I didn't see any way to set the minimum length. The ability to set the minimum length of a path would be sufficient for my application.

Rough idea for 2: For excluding self-crossing paths, my idea is to somehow tell sage "if you cross edge j while forming a path, do not use list of edges that cross j while creating the remainder of the path". Since the edge crossings in the underlying graph depends on how we draw it, I would think we have to manually tell sage which edges not to cross and when.

I am new to sage (and have no programming background), so I'm not sure how to do this, or if this would even work. Any help is really appreciated.

Example:

d = {1:[2,4,5], 2:[3,4,5,6], 3:[6], 4:[2,5], 5:[3,4,6]}
D = DiGraph(d)
D.all_simple_paths(starting_vertices=[1], ending_vertices=[6])


I'm drawing this graph as two squares, glued together along one edge, the edge 2--5, with an "X" inside each square (which is what sage output when I tried G=Graph(d), then show(G), and this is the embedding I had in mind).

This gives the following output:

[[1, 2, 6],
[1, 5, 6],
[1, 2, 3, 6],
[1, 2, 5, 6],
[1, 4, 2, 6],
[1, 4, 5, 6],
[1, 5, 3, 6],
[1, 2, 4, 5, 6],
[1, 2, 5, 3, 6],
[1, 4, 2, 3, 6],
[1, 4, 2, 5, 6],
[1, 4, 5, 3, 6],
[1, 5, 4, 2, 6],
[1, 2, 4, 5, 3, 6],
[1, 4, 2, 5, 3, 6],
[1, 5, 4, 2, 3, 6]]


First, I would like to filter this output so that only the maximal length paths are given:

[[1, 2, 4, 5, 3, 6],
[1, 4, 2, 5, 3, 6],
[1, 5, 4, 2, 3, 6]]


Finally, note that the last maximal path [1, 5, 4, 2, 3, 6] has a self-intersection (according to how we are drawing the underlying graph): the first edge 1-->5 will be crossed by the edge 4-->2 later in the path. As the first two maximal paths listed above do not have any self-intersections, the final output should be:

[[1, 2, 4, 5, 3, 6],
[1, 4, 2, 5, 3, 6]]


(edited for clarity and specificity, and added the example at the end)

### Maximal paths in all_simple_paths, and without edge crossings?

I am trying to use sage to list all simple paths between two fixed vertices of certain (embeddings of) graphs, but with these two additional conditions:

1. only paths of maximal length are output, and

2. of the maximal length paths from 1., only the paths without self-intersections are kept.

By self-intersection, I mean one edge of the path crossing another (which depends on the embedding of the underlying graph).

I can use all_simple_paths to output all simple paths from one vertex to the other by creating a dictionary d={} (specifying vertices, and the outgoing edges), then forming a digraph, then using all_simple_paths. But I'm not sure how to restrict the output list so that only simple paths both of maximal length and without self-crossings are given.

Rough idea for 1: I think you can set the maximum length of the paths all_simple_paths outputs, but I didn't see any way to set the minimum length. The ability to set the minimum length of a path would be sufficient for my application.

Rough idea for 2: For excluding self-crossing paths, my idea is to somehow tell sage "if you cross edge j while forming a path, do not use list of edges that cross j while creating the remainder of the path". Since the edge crossings in the underlying graph depends on how we draw it, I would think we have to manually tell sage which edges not to cross and when.

I am new to sage (and have no programming background), so I'm not sure how to do this, or if this would even work. Any help is really appreciated.

Example:

d = {1:[2,4,5], 2:[3,4,5,6], 3:[6], 4:[2,5], 5:[3,4,6]}
D = DiGraph(d)
D.all_simple_paths(starting_vertices=[1], ending_vertices=[6])


I'm drawing this graph as two squares, glued together along the edge 2--5, with an "X" inside each square (which is what sage output when I tried G=Graph(d), then show(G), and this is the embedding I had in mind).

This gives the following output:

[[1, 2, 6],
[1, 5, 6],
[1, 2, 3, 6],
[1, 2, 5, 6],
[1, 4, 2, 6],
[1, 4, 5, 6],
[1, 5, 3, 6],
[1, 2, 4, 5, 6],
[1, 2, 5, 3, 6],
[1, 4, 2, 3, 6],
[1, 4, 2, 5, 6],
[1, 4, 5, 3, 6],
[1, 5, 4, 2, 6],
[1, 2, 4, 5, 3, 6],
[1, 4, 2, 5, 3, 6],
[1, 5, 4, 2, 3, 6]]


First, I would like to filter this output so that only the maximal length paths are given:

[[1, 2, 4, 5, 3, 6],
[1, 4, 2, 5, 3, 6],
[1, 5, 4, 2, 3, 6]]


Finally, note that the last maximal path [1, 5, 4, 2, 3, 6] has a self-intersection (according to how we are drawing the underlying graph): the first edge 1-->5 will be crossed by the edge 4-->2 later in the path. As the first two maximal paths listed above do not have any self-intersections, the final output should be:

[[1, 2, 4, 5, 3, 6],
[1, 4, 2, 5, 3, 6]]


(edited for clarity and specificity, and added the example at the end)

### Maximal paths in all_simple_paths, and without edge crossings?

So far, I am trying to use sage using sage's all_simple_paths to list all simple paths between two fixed vertices of certain (embeddings of) graphs, but with these two additional conditions: graphs. I would like to go further, and restrict the output of all_simple_paths twice, to obtain the paths I'm interested in for the class of graphs I'm considering. Here are the successive restrictions I would like to implement:

1. only paths of maximal length the longest paths in the output of all_simple_paths are output, kept, and

2. of the maximal length longest paths from 1., only the paths without self-intersections are kept.

By self-intersection, I mean Here, a path P has a self-intersection if one edge of the path crossing P crosses over another (which edge of P (note that this definition depends on the embedding of the underlying graph).

I can use all_simple_paths to output all simple paths from one vertex to the other by creating a dictionary d={} (specifying vertices, and the outgoing edges), then forming a digraph, then using all_simple_paths. But I'm not sure how to restrict the output list so that only simple paths both of maximal length and without self-crossings are given.

Rough idea for 1: I think you can set the maximum length of the paths all_simple_paths outputs, but I didn't see any way to set the minimum length. The I think the ability to set the minimum length of a path would be sufficient for my application. application (but, being able to say "from the output of all_simple_paths, only keep the lists of length k" would also be useful to me).

Rough idea for 2: For excluding self-crossing After restricting the output of all_simple_paths the first time, which leaves only longest paths, my idea I'd like to do the following:

-Make a set, L, where the elements of this set are the longest paths just mentioned

-Write a function: Domain: L Range: {YES, NO} (would tell you if the path has a self-intersection).

The function would work as follows. Take the path l from L. This path is to somehow tell sage "if you cross edge j while forming a path, do not use list of a series of directed edges

i_1 --> i_2 --> ... i_m.

Now, we will consider each directed edge of the path l one-by-one, in order. So first, consider i_1 --> i_2. Call this edge e_1. Create a set C_{e_1} of all edges from the underlying graph that e_1 crosses, and remember this set until the function outputs YES or NO.

Now, move to the next edge in l, call it e_2. Check if e_2 is in C_{e_1}. If yes, halt and print NO. Else, create the set C_{e_2} of edges from the underlying graph that cross j while creating the remainder of the path". Since the edge crossings in the underlying graph depends e_2. Next, consider e_3. If e_3 is in the union of C_{e_1} and C_{e_2}, then halt and print NO. Else, create C_{e_3}, etc.

-Write a program which runs the function just described on how we draw it, I would think we have to manually tell sage which edges not to cross and when.

I am new to sage (and have no programming background), so I'm not sure how to do this, or if this would even work. Any help is really appreciated. each possible input in L, throws out all the "bad" paths, and returns only the "good" paths.

Example:

d = {1:[2,4,5], 2:[3,4,5,6], 3:[6], 4:[2,5], 5:[3,4,6]}
D = DiGraph(d)
D.all_simple_paths(starting_vertices=[1], ending_vertices=[6])


I'm drawing this graph as two squares, glued together along the edge 2--5, with an "X" inside each square (which is what sage output when I tried G=Graph(d), then show(G), and this is the embedding I had in mind).

This gives the following output:

[[1, 2, 6],
[1, 5, 6],
[1, 2, 3, 6],
[1, 2, 5, 6],
[1, 4, 2, 6],
[1, 4, 5, 6],
[1, 5, 3, 6],
[1, 2, 4, 5, 6],
[1, 2, 5, 3, 6],
[1, 4, 2, 3, 6],
[1, 4, 2, 5, 6],
[1, 4, 5, 3, 6],
[1, 5, 4, 2, 6],
[1, 2, 4, 5, 3, 6],
[1, 4, 2, 5, 3, 6],
[1, 5, 4, 2, 3, 6]]


First, I would like to filter this output so that only the maximal length longest paths are given:given (which forms the set "L"):

[[1, 2, 4, 5, 3, 6],
[1, 4, 2, 5, 3, 6],
[1, 5, 4, 2, 3, 6]]


Finally, note that the last maximal longest path [1, 5, 4, 2, 3, 6] is a "bad" path, since it has a self-intersection (according to how we are drawing the underlying graph): the first edge 1-->5 will be crossed by the edge 4-->2 later in the path. As the first two maximal longest paths listed above do not have any self-intersections, they are "good" paths, and the final output of the program should be:

[[1, 2, 4, 5, 3, 6],
[1, 4, 2, 5, 3, 6]]


### Maximal paths in all_simple_paths, and without edge crossings?

So far, I am using sage's all_simple_paths to list all simple paths between two fixed vertices of certain (embeddings of) graphs. I would like to go further, and restrict the output of all_simple_paths twice, to obtain the paths I'm interested in for the class of graphs I'm considering. Here are the successive restrictions I would like to implement:

1. only the longest paths in the output of all_simple_paths are kept, and

2. of the longest paths from 1., only the paths without self-intersections are kept.

Here, a path P has a self-intersection if one edge of P crosses over another edge of P (note that this definition depends on the embedding of the underlying graph).

Rough idea for 1: I think you can set the maximum length of the paths all_simple_paths outputs, but I didn't see any way to set the minimum length. I think the ability to set the minimum length of a path would be sufficient for my application (but, being able to say "from the output of all_simple_paths, only keep the lists of length k" would also be useful to me).

Rough idea for 2: After restricting the output of all_simple_paths the first time, which leaves only longest paths, I'd like to do the following:

-Make a set, L, where the elements of this set are the longest paths just mentioned

-Write a function: Domain: L Range: {YES, NO} (would tell you if the path has a self-intersection).

The function would work as follows. Take the path l from L. This path is a series of directed edges

i_1 --> i_2 --> ... i_m.

Now, we will consider each directed edge of the path l one-by-one, in order. So first, consider i_1 --> i_2. Call this edge e_1. Create a set C_{e_1} of all edges from the underlying graph that e_1 crosses, and remember this set until the function outputs YES or NO.

Now, move to the next edge in l, call it e_2. Check if e_2 is in C_{e_1}. If yes, halt and print NO. Else, create the set C_{e_2} of edges from the underlying graph that cross e_2. Next, consider e_3. If e_3 is in the union of C_{e_1} and C_{e_2}, then halt and print NO. Else, create C_{e_3}, etc.

-Write a program which runs the function just described on each possible input in L, throws out all the "bad" paths, and returns only the "good" paths.

Example:

d = {1:[2,4,5], 2:[3,4,5,6], 3:[6], 4:[2,5], 5:[3,4,6]}
D = DiGraph(d)
D.all_simple_paths(starting_vertices=[1], ending_vertices=[6])


I'm drawing this graph as two squares, glued together along the edge 2--5, with an "X" inside each square (which is what sage output when I tried G=Graph(d), then show(G), and this is the embedding I had in mind).

This gives the following output:

[[1, 2, 6],
[1, 5, 6],
[1, 2, 3, 6],
[1, 2, 5, 6],
[1, 4, 2, 6],
[1, 4, 5, 6],
[1, 5, 3, 6],
[1, 2, 4, 5, 6],
[1, 2, 5, 3, 6],
[1, 4, 2, 3, 6],
[1, 4, 2, 5, 6],
[1, 4, 5, 3, 6],
[1, 5, 4, 2, 6],
[1, 2, 4, 5, 3, 6],
[1, 4, 2, 5, 3, 6],
[1, 5, 4, 2, 3, 6]]


First, I would like to filter this output so that only the longest paths are given (which forms the set "L"):

[[1, 2, 4, 5, 3, 6],
[1, 4, 2, 5, 3, 6],
[1, 5, 4, 2, 3, 6]]


Finally, note that the last longest path [1, 5, 4, 2, 3, 6] is a "bad" path, since it has a self-intersection (according to how we are drawing the underlying graph): the first edge 1-->5 will be crossed by the edge 4-->2 later in the path. As the first two longest paths listed above do not have any self-intersections, they are "good" paths, and the final output of the program should be:

[[1, 2, 4, 5, 3, 6],
[1, 4, 2, 5, 3, 6]]


### Maximal paths in Restrict output of all_simple_paths, and without edge crossings?to obtain longest simple paths that do not have self-intersections?

So far, I am using sage's all_simple_paths to list all simple paths between two fixed vertices of certain (embeddings of) graphs. I would like to go further, and restrict the output of all_simple_paths twice, to obtain the paths I'm interested in for the class of graphs I'm considering. Here are the successive restrictions I would like to implement:

1. only the longest paths in the output of all_simple_paths are kept, and

2. of the longest paths from 1., only the paths without self-intersections are kept.

Here, a path P has a self-intersection if one edge of P crosses over another edge of P (note that this definition depends on the embedding of the underlying graph).

Rough idea for 1: I think you can set the maximum length of the paths all_simple_paths outputs, but I didn't see any way to set the minimum length. I think the ability to set the minimum length of a path would be sufficient for my application (but, being able to say "from the output of all_simple_paths, only keep the lists of length k" would also be useful to me).

Rough idea for 2: After restricting the output of all_simple_paths the first time, which leaves only longest paths, I'd like to do the following:

-Make a set, L, where the elements of this set are the longest paths just mentioned

-Write a function: Domain: L Range: {YES, NO} (would tell you if the path has a self-intersection).

The function would work as follows. Take the path l from L. This path is a series of directed edges

i_1 --> i_2 --> ... i_m.

Now, we will consider each directed edge of the path l one-by-one, in order. So first, consider i_1 --> i_2. Call this edge e_1. Create a set C_{e_1} of all edges from the underlying graph that e_1 crosses, and remember this set until the function outputs YES or NO.

Now, move to the next edge in l, call it e_2. Check if e_2 is in C_{e_1}. If yes, halt and print NO. Else, create the set C_{e_2} of edges from the underlying graph that cross e_2. Next, consider e_3. If e_3 is in the union of C_{e_1} and C_{e_2}, then halt and print NO. Else, create C_{e_3}, etc.

-Write a program which runs the function just described on each possible input in L, throws out all the "bad" paths, and returns only the "good" paths.

Example:

d = {1:[2,4,5], 2:[3,4,5,6], 3:[6], 4:[2,5], 5:[3,4,6]}
D = DiGraph(d)
D.all_simple_paths(starting_vertices=[1], ending_vertices=[6])


I'm drawing this graph as two squares, glued together along the edge 2--5, with an "X" inside each square (which is what sage output when I tried G=Graph(d), then show(G), and this is the embedding I had in mind).

This gives the following output:

[[1, 2, 6],
[1, 5, 6],
[1, 2, 3, 6],
[1, 2, 5, 6],
[1, 4, 2, 6],
[1, 4, 5, 6],
[1, 5, 3, 6],
[1, 2, 4, 5, 6],
[1, 2, 5, 3, 6],
[1, 4, 2, 3, 6],
[1, 4, 2, 5, 6],
[1, 4, 5, 3, 6],
[1, 5, 4, 2, 6],
[1, 2, 4, 5, 3, 6],
[1, 4, 2, 5, 3, 6],
[1, 5, 4, 2, 3, 6]]


First, I would like to filter this output so that only the longest paths are given (which forms the set "L"):

[[1, 2, 4, 5, 3, 6],
[1, 4, 2, 5, 3, 6],
[1, 5, 4, 2, 3, 6]]


Finally, note that the last longest path [1, 5, 4, 2, 3, 6] is a "bad" path, since it has a self-intersection (according to how we are drawing the underlying graph): the first edge 1-->5 will be crossed by the edge 4-->2 later in the path. As the first two longest paths listed above do not have any self-intersections, they are "good" paths, and the final output of the program should be:

[[1, 2, 4, 5, 3, 6],
[1, 4, 2, 5, 3, 6]]


### Restrict output of all_simple_paths, to obtain all longest simple paths that do not have edge self-intersections?

So far, I am using sage's all_simple_paths to list all simple paths between two fixed vertices of certain (embeddings of) graphs. I would like to go further, and restrict the output of all_simple_paths twice, to obtain the paths I'm interested in for the class of graphs I'm considering. Here are the successive restrictions I would like to implement:

1. only the longest paths in the output of all_simple_paths are kept, and

2. of the longest paths from 1., only the paths without edge self-intersections are kept.

Here, a path P has a an edge self-intersection if one edge of P crosses over another edge of P (note that this definition depends on the embedding of the underlying graph).

Rough idea for 1: I think you can set the maximum length of the paths all_simple_paths outputs, but I didn't see any way to set the minimum length. I think the ability to set the minimum discard all paths less than or equal to a certain length of a path from the output of all_simple_paths would be sufficient for my application (but, being able to say "from the output of all_simple_paths, only keep the lists of length k" would also be useful to me).

application.

Rough idea for 2: After restricting the output of all_simple_paths the first time, which leaves only the longest simple paths, I'd like to do the following:

-Make a set, L, where the elements of this set are the longest simple paths just mentioned

-Write a function: Domain: L Range: {YES, NO} (would tell you if the path has a self-intersection).

The function would work as follows. following is the rule defining the function. Take the path l from L. This path is a series of directed edges

i_1 --> i_2 --> ... i_m.

Now, we will consider each directed edge of the path l one-by-one, in order. So first, consider i_1 --> i_2. Call this edge e_1. Create a set C_{e_1} of all edges from the underlying graph that e_1 crosses, and remember this set until the function outputs YES or NO.

Now, move to the next edge in l, call it e_2. Check if e_2 is in C_{e_1}. If yes, halt and print NO. Else, create the set C_{e_2} of edges from the underlying graph that cross e_2. e_2, and remember this set until the function outputs YES or NO. Next, consider e_3. If e_3 is in the union of C_{e_1} and C_{e_2}, then halt and print NO. Else, create C_{e_3}, etc.

-Write a program which runs the function just described on each possible input in L, then throws out all the "bad" NO paths, and returns only the "good" YES paths.

Example:

d = {1:[2,4,5], 2:[3,4,5,6], 3:[6], 4:[2,5], 5:[3,4,6]}
D = DiGraph(d)
D.all_simple_paths(starting_vertices=[1], ending_vertices=[6])


I'm drawing this graph as two squares, glued together along the edge 2--5, with an "X" inside each square (which is what sage output when I tried G=Graph(d), then show(G), and this is the embedding I had in mind).

This gives the following output:

[[1, 2, 6],
[1, 5, 6],
[1, 2, 3, 6],
[1, 2, 5, 6],
[1, 4, 2, 6],
[1, 4, 5, 6],
[1, 5, 3, 6],
[1, 2, 4, 5, 6],
[1, 2, 5, 3, 6],
[1, 4, 2, 3, 6],
[1, 4, 2, 5, 6],
[1, 4, 5, 3, 6],
[1, 5, 4, 2, 6],
[1, 2, 4, 5, 3, 6],
[1, 4, 2, 5, 3, 6],
[1, 5, 4, 2, 3, 6]]


First, I would like to filter this output so that only the longest simple paths are given (which forms the set "L"):

[[1, 2, 4, 5, 3, 6],
[1, 4, 2, 5, 3, 6],
[1, 5, 4, 2, 3, 6]]


Finally, note that the last longest path [1, 5, 4, 2, 3, 6] is a "bad" NO path, since it has a an edge self-intersection (according to how we are drawing the underlying graph): the first edge 1-->5 will be crossed by the edge 4-->2 later in the path. As the first two longest simple paths listed above do not have any edge self-intersections, they are "good" YES paths, and the final output of the program should be:

[[1, 2, 4, 5, 3, 6],
[1, 4, 2, 5, 3, 6]]


### Restrict output of all_simple_paths, to obtain all longest simple paths that do not have edge self-intersections?

So far, I am using sage's all_simple_paths to list all simple paths between two fixed vertices of certain (embeddings of) graphs. I would like to go further, and restrict the output of all_simple_paths twice, to obtain the paths I'm interested in for the class of graphs I'm considering. Here are the successive restrictions I would like to implement:

1. only the longest paths in the output of all_simple_paths are kept, and

2. of the longest paths from 1., only the paths without edge self-intersections are kept.

Here, a path P has an edge self-intersection if one edge of P crosses over another edge of P (note that this definition depends on the embedding of the underlying graph).

Rough idea for 1: I think you can set the maximum length of the paths all_simple_paths outputs, but I didn't see any way to set the minimum length. I think the ability to discard all paths less than or equal to a certain length from the output of all_simple_paths would be sufficient for my application.

Rough idea for 2: After restricting the output of all_simple_paths the first time, which leaves only the longest simple paths, I'd like to do the following:

-Make a set, L, where the elements of this set are the longest simple paths just mentioned

-Write a function: Domain: L Range: {YES, NO} (would tell you if the path has a self-intersection).

The following is the rule defining the function. Take the path l from L. This path is a series of directed edges

i_1 --> i_2 --> ... i_m.

Now, we will consider each directed edge of the path l one-by-one, in order. So first, consider i_1 --> i_2. Call this edge e_1. Create a set C_{e_1} of all edges from the underlying graph that e_1 crosses, and remember this set until the function outputs YES or NO.

Now, move to the next edge in l, call it e_2. Check if e_2 is in C_{e_1}. If yes, halt and print NO. Else, create the set C_{e_2} of edges from the underlying graph that cross e_2, and remember this set until the function outputs YES or NO. Next, consider e_3. If e_3 is in the union of C_{e_1} and C_{e_2}, then halt and print NO. Else, create C_{e_3}, etc.

-Write a program which runs the function just described on each possible input in L, then throws out all the NO paths, and returns only the YES paths.

Example:

d = {1:[2,4,5], 2:[3,4,5,6], 3:[6], 4:[2,5], 5:[3,4,6]}
D = DiGraph(d)
D.all_simple_paths(starting_vertices=[1], ending_vertices=[6])


I'm drawing this graph as two squares, glued together along the edge 2--5, with an "X" inside each square (which is what sage output when I tried G=Graph(d), then show(G), and this is the embedding I had in mind).

This gives the following output:

[[1, 2, 6],
[1, 5, 6],
[1, 2, 3, 6],
[1, 2, 5, 6],
[1, 4, 2, 6],
[1, 4, 5, 6],
[1, 5, 3, 6],
[1, 2, 4, 5, 6],
[1, 2, 5, 3, 6],
[1, 4, 2, 3, 6],
[1, 4, 2, 5, 6],
[1, 4, 5, 3, 6],
[1, 5, 4, 2, 6],
[1, 2, 4, 5, 3, 6],
[1, 4, 2, 5, 3, 6],
[1, 5, 4, 2, 3, 6]]


First, I would like to filter this output so that only the longest simple paths are given (which forms the set "L"):

[[1, 2, 4, 5, 3, 6],
[1, 4, 2, 5, 3, 6],
[1, 5, 4, 2, 3, 6]]


Finally, note that the last longest path [1, 5, 4, 2, 3, 6] is a NO path, since it has an edge self-intersection (according to how we are drawing the underlying graph): the first edge 1-->5 will be crossed by the edge 4-->2 later in the path. As the first two longest simple paths listed above do not have any edge self-intersections, they are YES paths, and the final output of the program should be:

[[1, 2, 4, 5, 3, 6],
[1, 4, 2, 5, 3, 6]]


### Restrict output of all_simple_paths, to obtain all only longest simple paths that do not have edge self-intersections?

So far, I am using sage's all_simple_paths to list all simple paths between two fixed vertices of certain (embeddings of) graphs. I would like to go further, and restrict the output of all_simple_paths twice, to obtain the paths I'm interested in for the class of graphs I'm considering. Here are the successive restrictions I would like to implement:

1. only the longest paths in the output of all_simple_paths are kept, kept (EDIT: this can be done with list comprehension, see example below), and

2. of the longest paths from 1., only the paths without edge self-intersections are kept.

Here, a path P has an edge self-intersection if one edge of P crosses over another edge of P (note that this definition depends on the embedding of the underlying graph).

Rough idea for 1: I think you can set the maximum length of the paths all_simple_paths outputs, but I didn't see any way to set the minimum length. I think the ability to discard all paths less than or equal to a certain length from the output of all_simple_paths would be sufficient for my application.application (EDIT: this can be done with list comprehension, see example below).

Rough idea for 2: After restricting the output of all_simple_paths the first time, which leaves only the longest simple paths, I'd like to do the following:

-Make a set, L, where the elements of this set are the longest simple paths just mentioned

-Write a function: Domain: L Range: {YES, NO} (would tell you if the path has a self-intersection).

The following is the rule defining the function. Take the path l from L. This path is a series of directed edges

i_1 --> i_2 --> ... i_m.

Now, we will consider each directed edge of the path l one-by-one, in order. So first, consider i_1 --> i_2. Call this edge e_1. Create a set C_{e_1} of all edges from the underlying graph that e_1 crosses, and remember this set until the function outputs YES or NO.

Now, move to the next edge in l, call it e_2. Check if e_2 is in C_{e_1}. If yes, halt and print NO. Else, create the set C_{e_2} of edges from the underlying graph that cross e_2, and remember this set until the function outputs YES or NO. Next, consider e_3. If e_3 is in the union of C_{e_1} and C_{e_2}, then halt and print NO. Else, create C_{e_3}, etc.

-Write a program which runs the function just described on each possible input in L, then throws out all the NO paths, and returns only the YES paths.

Example:

d = {1:[2,4,5], 2:[3,4,5,6], 3:[6], 4:[2,5], 5:[3,4,6]}
D = DiGraph(d)
D.all_simple_paths(starting_vertices=[1], ending_vertices=[6])


I'm drawing this graph as two squares, glued together along the edge 2--5, with an "X" inside each square (which is what sage output when I tried G=Graph(d), then show(G), and this is the embedding I had in mind).

This gives the following output:

[[1, 2, 6],
[1, 5, 6],
[1, 2, 3, 6],
[1, 2, 5, 6],
[1, 4, 2, 6],
[1, 4, 5, 6],
[1, 5, 3, 6],
[1, 2, 4, 5, 6],
[1, 2, 5, 3, 6],
[1, 4, 2, 3, 6],
[1, 4, 2, 5, 6],
[1, 4, 5, 3, 6],
[1, 5, 4, 2, 6],
[1, 2, 4, 5, 3, 6],
[1, 4, 2, 5, 3, 6],
[1, 5, 4, 2, 3, 6]]


First, I would like to filter this output so that only the longest simple paths are given (which forms the set "L"):"L"). EDIT: This can be done with list comprehension as follows. Let F_1 = D.all_simple_paths(starting_vertices=[1], ending_vertices=[6]). Then using [x for x in F_1 if len(x) > 5] leaves only longest simple paths:

[[1, 2, 4, 5, 3, 6],
[1, 4, 2, 5, 3, 6],
[1, 5, 4, 2, 3, 6]]


Finally, note that the last longest path [1, 5, 4, 2, 3, 6] is a NO path, since it has an edge self-intersection (according to how we are drawing the underlying graph): the first edge 1-->5 will be crossed by the edge 4-->2 later in the path. As the first two longest simple paths listed above do not have any edge self-intersections, they are YES paths, and the final output of the program should be:

[[1, 2, 4, 5, 3, 6],
[1, 4, 2, 5, 3, 6]]