# Edges and vertices relationship problems

### Graph theory - Wikipedia

Graph theory deals with routing and network problems and if it is possible to find a. “best” route, whether A network is a connection of vertices through edges. His work on the “Königsberg Bridge Problem” is considered by many to be the beginning of . of its vertex set, and the size of a graph is the cardinality of its edge set. .. every possible connection of a vertex of X with a vertex of Y is present in. Problem description: What is the smallest subset of vertices (edges) whose deletion (vertex) deletions sufficient to disconnect G. There is a close relationship.

Not every regular graph has a 1-factorization; for instance, the Petersen graph does not.

## Euler's polyhedron formula

More generally the snarks are defined as the graphs that, like the Petersen graph, are bridgeless, 3-regular, and of class 2. The theorem was stated earlier in terms of projective configurations and was proven by Ernst Steinitz. In a result that inspired Vizing, [10] Shannon showed that this is the worst case: Algorithms[ edit ] Because the problem of testing whether a graph is class 1 is NP-completethere is no known polynomial time algorithm for edge-coloring every graph with an optimal number of colors.

Nevertheless, a number of algorithms have been developed that relax one or more of these criteria: The algorithm of Alon begins by making the input graph regular, without increasing its degree or significantly increasing its size, by merging pairs of vertices that belong to the same side of the bipartition and then adding a small number of additional vertices and edges.

Then, if the degree is odd, Alon finds a single perfect matching in near-linear time, assigns it a color, and removes it from the graph, causing the degree to become even. Finally, Alon applies an observation of Gabowthat selecting alternating subsets of edges in an Euler tour of the graph partitions it into two regular subgraphs, to split the edge coloring problem into two smaller subproblems, and his algorithm solves the two subproblems recursively.

The total time for his algorithm is O m log m. Make the input multigraph G Eulerian by adding a new vertex connected by an edge to every odd-degree vertex, find an Euler tour, and choose an orientation for the tour. Form a bipartite graph H in which there are two copies of each vertex of G, one on each side of the bipartition, with an edge from a vertex u on the left side of the bipartition to a vertex v on the right side of the bipartition whenever the oriented tour has an edge from u to v in G.

Apply a bipartite graph edge coloring algorithm to H. Each color class in H corresponds to a set of edges in G that form a subgraph with maximum degree two; that is, a disjoint union of paths and cycles, so for each color class in H it is possible to form three color classes in G.

Next, count the number of edges the polyhedron has, and call this number E. Finally, count the number of faces and call it F. Euler's formula is true for the cube and the icosahedron. It turns out, rather beautifully, that it is true for pretty much every polyhedron. The only polyhedra for which it doesn't work are those that have holes running through them like the one shown in the figure below.

## Graph theory

This polyhedron has a hole running through it. Euler's formula does not hold in this case. These polyhedra are called non-simple, in contrast to the ones that don't have holes, which are called simple. Non-simple polyhedra might not be the first to spring to mind, but there are many of them out there, and we can't get away from the fact that Euler's Formula doesn't work for any of them. However, even this awkward fact has become part of a whole new theory about space and shape.

The power of Euler's formula Whenever mathematicians hit on an invariant feature, a property that is true for a whole class of objects, they know that they're onto something good. They use it to investigate what properties an individual object can have and to identify properties that all of them must have.

Euler's formula can tell us, for example, that there is no simple polyhedron with exactly seven edges. You don't have to sit down with cardboard, scissors and glue to find this out — the formula is all you need.

The argument showing that there is no seven-edged polyhedron is quite simple, so have a look at it if you're interested. Using Euler's formula in a similar way we can discover that there is no simple polyhedron with ten faces and seventeen vertices.

The prism shown below, which has an octagon as its base, does have ten faces, but the number of vertices here is sixteen. The pyramid, which has a 9-sided base, also has ten faces, but has ten vertices. But Euler's formula tells us that no simple polyhedron has exactly ten faces and seventeen vertices.

Both these polyhedra have ten faces, but neither has seventeen vertices. It's considerations like these that lead us to what's probably the most beautiful discovery of all. It involves the Platonic Solids, a well-known class of polyhedra named after the ancient Greek philosopher Platoin whose writings they first appeared.

From left to right we have the tetrahedon with four faces, the cube with six faces, the octahedron with eight faces, the dodecahedron with twelve faces, and the icosahedron with twenty faces.

Although their symmetric elegance is immediately apparent when you look at the examples above, it's not actually that easy to pin it down in words. It turns out that it is described by two features. The first is that Platonic solids have no spikes or dips in them, so their shape is nice and rounded. In other words, this means that whenever you choose two points in a Platonic solid and draw a straight line between them, this piece of straight line will be completely contained within the solid — a Platonic solid is what is called convex.

The second feature, called regularity, is that all the solid's faces are regular polygons with exactly the same number of sides, and that the same number of edges come out of each vertex of the solid.

The cube is regular, since all its faces are squares and exactly three edges come out of each vertex.

### Edge coloring - Wikipedia

You can verify for yourself that the tetrahedron, the octahedron, the icosahedron and the dodecahedron are also regular. Now, you might wonder how many different Platonic Solids there are. Ever since the discovery of the cube and tetrahedron, mathematicians were so attracted by the elegance and symmetry of the Platonic Solids that they searched for more, and attempted to list all of them.

This is where Euler's formula comes in. In a weakly connected graph, there may exist paths to nodes from which there is no way to return. What if I want to split the graph into equal-sized pieces? For example, suppose we want to break a computer program spread across several files into two maintainable units.

Construct a graph where the vertices are subroutines, with an edge between any two subroutines that interact, say by one calling the other. We seek to partition the routines into equal-sized sets so that the fewest pairs of interacting routines are spread across the set.

This is the graph partition problem, which is further discussed in Section. Although the problem is NP-complete, reasonable heuristics exist to solve it.

### Euler's polyhedron formula | knifedirectory.info

Is there one weak link in my graph? Any vertex that is such a weak point is called an articulation vertex. A bridge is the analogous concept for edges, meaning a single edge whose deletion disconnects the graph. The simplest algorithms for identifying articulation vertices or bridges would try deleting vertices or edges one by one, and then using DFS or BFS to test whether the resulting graph is still connected.

More complicated but linear-time algorithms exist for both problems, based on depth-first search. Implementations are described below and in Section.