question: why a graph can only have 2 horizontal Asymtotes.
comment: On a normal 2 axis (ie 2D) graph you can be 3 things, above the line, below the line, and on the line. An asymptote by definition means always above or below the line (as the line never quite reaches it even though it gets real close), so an asymptote can only be above or below the line. Since there are only two options, you can only have two asymptotes. You can't have a third asymptote because there's nowhere to put it. Above the line is covered, below the line is covered, where would the third one go? However, you can have a third asymptote if you are using a 3 axis (ie 3D) graph as that adds a third dimension for you to work with.
passages: These rules are meant to hold for all configurations which are congruent to the ones shown in the figure. In other words, the figures involved in the relations can be in all possible positions and orientations. The labels on the horizontal edges are simply auxiliary variables introduced to make the formulas simpler. Once a single row of non-horizontal edges is provided, the remaining rows are uniquely determined by the compatibility rules. 
  
  Section::::Invariant structures.
  
  Section::::Invariant structures.:Corner coordinate products.
  
  It follows directly from the formula for the pentagram map, in terms of corner coordinates, that the two quantities
   BULLET::::2. Replace any vertex of the graph by a pair of vertices, each of which has the same set of neighbors as the replaced vertex. The new pair of vertices are called "false twins" of each other.
  
  BULLET::::3. Replace any vertex of the graph by a pair of vertices, each of which has as its neighbors the neighbors of the replaced vertex together with the other vertex of the pair. The new pair of vertices are called "true twins" of each other.
   BULLET::::- A horizontal or vertical arrow represents an indel. Horizontal arrows will align a gap ("-") to the letter of the row (the "side" sequence), vertical arrows will align a gap to the letter of the column (the "top" sequence).
  
  BULLET::::- If there are multiple arrows to choose from, they represent a branching of the alignments. If two or more branches all belong to paths from the bottom left to the top right cell, they are equally viable alignments. In this case, note the paths as separate alignment candidates.
   Quartet decomposition
   The "signature similarity" is computed as follows. For a node "u" in graph "G", "u" denotes the "i" coordinate of its signature vector, i.e., "u" is the number of times node "u" is touched by an orbit "i" in "G". The distance "D(u,v)" between the "i" orbits of nodes "u" and "v" is defined as: 
  
  formula_15, 
  
  where "w" is the weight of orbit "i" that accounts for dependencies between orbits (see [Milenković and Pržulj, 2008] for details). The total distance "D(u,v)" between nodes "u" and "v" is defined as: 
  
  formula_16. 
   As well as "K" and "K", there are some other exceptional small graphs with the property that their line graph has a higher degree of symmetry than the graph itself. For instance, the diamond graph "K" (two triangles sharing an edge) has four graph automorphisms but its line graph "K" has eight. In the illustration of the diamond graph shown, rotating the graph by 90 degrees is not a symmetry of the graph, but is a symmetry of its line graph. However, all such exceptional cases have at most four vertices. A strengthened version of the Whitney isomorphism theorem states that, for connected graphs with more than four vertices, there is a one-to-one correspondence between isomorphisms of the graphs and isomorphisms of their line graphs.
   For each orbit "j", one needs to measure the "j" GDD, "d(k)", i.e., the distribution of the number of nodes in "G" "touching" the corresponding graphlet at orbit "j" "k" times. Clearly, the degree distribution is the "0th" GDD. "d(k)" is scaled as
  
  formula_7 to decrease the contribution of larger degrees in a GDD and then normalized with respect to its total area
  
  formula_8 giving the "normalized distribution"
  
  formula_9.
  
  The "j" GDD-agreement compares the "j" GDDs of two networks.
  
  For two networks "G" and "H" and a particular orbit "j", the "distance" "D(G,H)" between their normalized "j" GDDs is: 
   In the case of formula_50 the equation is formula_51 and the set of its solutions formula_52 has a vertical line as its graph, as shown in the figure to the right. The value formula_53 where the line intersects the formula_12-axis in the point formula_55, is called an  Except for formula_56 when the graph coincides with the graphs of this kind do not intersect the they have no 
   BULLET::::- The set of pairs formula_17 used to define the graph formula_20 may also include points where one climber is at a peak and the other is at a valley. These points may be interpreted as isolated vertices of formula_20: neither climber can move, so the degree is zero.
  
  According to the handshaking lemma, every connected component of an undirected graph has an even number of odd-degree vertices.
   BULLET::::- By increasing the size of the "k"-mers, the number of vertices will also decrease. This will help with the construction of the genome as there will be fewer paths to traverse in the graph.
  
  BULLET::::- Larger "k"-mers also run a higher risk of not having outward vertices from every k-mer. This is due to larger "k"-mers increasing the risk that it will not overlap with another "k"-mer by formula_10. Therefore, this can lead to disjoints in the reads, and as such, can lead to a higher amount of smaller contigs.
   And h = V (column corresponding to the smallest singular vector). This is true since h lies in the null space of A. Since we have 8 degrees of freedom the algorithm requires at least four point correspondences.
  
  In case when RANSAC is used to estimate the homography and multiple correspondences are available the correct homography matrix is the one with the maximum number of inliers.
  
  Section::::Process stages.:Calibration.:Compositing.
   BULLET::::4. Klumpenhouwer Networks (a) and (b), will be isographic under the circumstance each T-number of Network (b) is 7 times the corresponding T-number in Network (a), and each I-number of Network (b) is exactly j more than 7 times the corresponding I-number in Network (a)...F(7,j): F(7,j)(T)=T; F(7,j)(I)=I.
  
  BULLET::::5. "Klumpenhouwer Networks (a) and (b), even if sharing the same configuration of nodes and arrows, will not be isographic under any other circumstances."
   BULLET::::- The shortest path distances obey Ptolemy's inequality: for every four vertices , , , and , the inequality holds. For instance, the gem graph (3-fan) in the illustration is not Ptolemaic, because in this graph , greater than .
  
  BULLET::::- For every two overlapping maximal cliques, the intersection of the two cliques is a separator that splits the differences of the two cliques. In the illustration of the gem graph, this is not true: cliques and are not separated by their intersection, , because there is an edge that connects the cliques but avoids the intersection.
   BULLET::::2. If "X" and "X" both contain a vertex "v", then all nodes "X" of "T" in the (unique) path between "X" and "X" contain "v" as well. Equivalently, the tree nodes containing vertex "v" form a connected subtree of "T".
  
  BULLET::::3. For every edge ("v", "w") in the graph, there is a subset "X" that contains both "v" and "w". That is, vertices are adjacent in the graph only when the corresponding subtrees have a node in common.
   In the pathfinder network, the entities correspond to the nodes of the generated network, and the links in the network are determined by the patterns of proximities. For example, if the proximities are similarities, links will generally connect nodes of high similarity. The links in the network will be undirected if the proximities are symmetrical for every pair of entities. Symmetrical proximities mean that the order of the entities is not important, so the proximity of "i" and "j" is the same as the proximity of "j" and "i" for all pairs "i,j". If the proximities are not symmetrical for every pair, the links will be directed.
   BULLET::::- Having smaller sizes will increase the chance for all the "k"-mers to overlap, and as such, have the required subsequences in order to construct the De Bruijn graph.
  
  BULLET::::- However, by having smaller sized "k"-mers, you also risk having many vertices in the graph leading into a single k-mer. Therefore, this will make the reconstruction of the genome more difficult as there is a higher level of path ambiguities due to the larger amount of vertices that will need to be traversed.
  
  BULLET::::- Information is lost as the "k"-mers become smaller.
   If the network structure of the model is a directed acyclic graph, the model represents a factorization of the joint probability of all random variables. More precisely, if the events are formula_1 then the joint probability satisfies
  
  where formula_3 is the set of parents of node formula_4 (nodes with edges directed towards formula_4). In other words, the joint distribution factors into a product of conditional distributions. For example, the graphical model in the Figure shown above (which is actually not a directed acyclic graph, but an ancestral graph) consists of the random variables formula_6
   BULLET::::- The graph can be constructed from a single vertex by a sequence of operations that add a new degree-one (pendant) vertex, or duplicate (twin) an existing vertex, with the exception that a twin operation in which the new duplicate vertex is not adjacent to its twin (false twins) can only be applied when the neighbors of the twins form a clique. These three operations without the exception form all distance-hereditary graph. To form all Ptolemaic graphs, it is not enough to use pendant vertices and true twins; the exceptional case of false twins is sometimes also required.
   "To generate isomorphic graphs, the mapping f must be what is called an "automorphism" of the T/I system. Networks that have isomorphic graphs are called isographic."
  
  "to be isographic, two networks must have these features:
  
  BULLET::::1. They must have the same configuration of nodes and arrows.
  
  BULLET::::2. There must be some isomorphism F that maps the transformation-system used to label the arrows of one network, into the transformation-system used to label the arrows of the other.
  
  BULLET::::3. If the transformation X labels an arrow of the one network, then the transformation F(X) labels the corresponding arrow of the other."
   As well, researchers have studied efficient algorithms for constructing smaller portions of an arrangement, such as zones, "k"-levels, or the set of cells containing a given set of points. The problem of finding the arrangement vertex with the median "x"-coordinate arises (in a dual form) in robust statistics as the problem of computing the Theil–Sen estimator of a set of points.
   If we sum every 6th term of the second row, we find that in every case it is equal to a+b+c+d+e+f. Since every term is the sum of the six above it and the terms we are concerned with are six apart, there is no overlap. It is not difficult to see this holds true for every row of the 6-nomial triangle (we are interested in the nth row specifically) and m-nomial triangles as well.
   BULLET::::- A "k"-crossing "h"-layer drawing of a graph "G" is a placement of the vertices of "G" onto "h" distinct horizontal lines, with edges routed as monotonic polygonal paths between these lines, in such a way that there are at most "k" crossings. The graphs with such drawings have pathwidth that is bounded by a function of "h" and "k". Therefore, when "h" and "k" are both constant, it is possible in linear time to determine whether a graph has a "k"-crossing "h"-layer drawing.
   BULLET::::- "Signed graph models": Every path in a signed graph has a sign from the product of the signs on the edges. Under the assumptions of balance theory, edges may change sign and result in a bifurcated graph. The weaker "clusterability axiom" (no cycle has exactly one negative edge) yields results with more than two clusters, or subgraphs with only positive edges.
   The foldings of finite diagrams are:
  
  BULLET::::- formula_48
  
  BULLET::::- formula_49
  
  BULLET::::- formula_50 (if quotienting by the full group or a 3-cycle, in addition to formula_51 in 3 different ways, if quotienting by an involution)
  
  BULLET::::- formula_52
  
  Similar foldings exist for affine diagrams, including:
  
  BULLET::::- formula_53
  
  BULLET::::- formula_54
  
  BULLET::::- formula_55
  
  BULLET::::- formula_56
   BULLET::::9. Symbols connected to their neighbors by line segments form text lines. Using all the centroids in a text line, one can compute an actual line segment representing the text line with linear regression. This is important because it is unlikely that all the centroids of symbols in a text line are actually collinear.
  
  BULLET::::10. For each pair of text lines, one can compute a minimum distance between their corresponding line segments. If this distance is within some tolerance of the between-line spacing calculated in step 7, then the two text lines are grouped into the same text block.
  