Graphs:Eulerian and Hamiltonian Graphs

By -

Eulerian and Hamiltonian Graphs

A path when generalized to include visiting a vertex more than once is called a trail. In other words, a trail is a sequence of edges (v1, v2), (v2, v3),.. ., (vk−2 , vk−1), (vk−1 , vk) in which all the vertices (v1, v2,... , vk ) may not be distinct but all the edges are distinct. Sometimes a trail is referred to as a (non-simple) path and path is referred to as a simple path.

For example in Figure 4.8(a) (b, a), (a, c), (c, d), (d, a), (a, f ) is a trail (but not a simple path because vertex a is visited twice.

If the first and the last vertex in a trail are the same, it is called a closed trail , otherwise an open trail . An Eulerian trail in a graph G = (V, E) is one that includes every edge in E (exactly once). A graph with a closed Eulerian trail is called a Eulerian graph. Equivalently, in an Eulerian graph, G, starting from a vertex one can traverse every edge in G exactly once and return to the starting vertex. According to a theorem proved by Euler in 1736, (considered the beginning of graph theory), a connected graph is Eulerian if and only if the degree of its every vertex is even.

Given a connected graph G it is easy to check if G is Eulerian. Finding an actual Eulerian trail of G is more involved. An efficient algorithm for traversing the edges of G to obtain an Euler trail was given by Fleury. The details can be found in [20].

A cycle in a graph G is said to be Hamiltonian if it passes through every vertex of G. Many families of special graphs are known to be Hamiltonian, and a large number of theorems have been proved that give sufficient conditions for a graph to be Hamiltonian.

However, the problem of determining if an arbitrary graph is Hamiltonian is NP-complete.

Graph theory, a branch of combinatorial mathematics, has been studied for over two centuries. However, its applications and algorithmic aspects have made enormous advances only in the past fifty years with the growth of computer technology and operations research.

Here we have discussed just a few of the better-known problems and algorithms. Additional material is available in the references provided. In particular, for further exploration the Stanford GraphBase [10], the LEDA [12], and the Graph Boost Library [17] provide valuable and interesting platforms with collection of graph-processing programs and benchmark databases.

Comments

Post a Comment

Popular posts from this blog

0/1 Knapsack Problem Memory function.

By -
Image
0 / 1 Knapsack Problem Memory function Definition: Given a set of n items of known weights w1,…,wn and values v1,…,vn and a knapsack of capacity W, the problem is to find the most valuable subset of the items that fit into the knapsack. Knapsack problem is an OPTIMIZATION PROBLEM Dynamic programming approach to solve knapsack problem Step 1: Identify the smaller sub-problems. If items are labeled 1..n, then a sub-problem would be to find an optimal solution for Sk = {items labeled 1, 2, .. k } Step 2: Recursively define the value of an optimal solution in terms of solutions to smaller problems. Initial conditions: S te p 3: Bottom up computation using iteration Question: Apply bottom-up dynamic programming algorithm to the following instance of the knapsack problem Capacity W= 5 Solution: Using dynamic programming approach, we have: What is the composition of the optimal subset? The composition of the optimal subset if found by tracing back the computations for the...
Read more

Binary Space Partitioning Trees:BSP Tree as a Hierarchy of Regions.

By -
Image
BSP Tree as a Hierarchy of Regions Another way to look at BSP Trees is to focus on the hierarchy of regions created by the recursive partitioning, instead of focusing on the hyperplanes themselves. This view helps us to see more easily how intersections are efficiently computed. The key idea is to think of a BSP Tree region as serving as a bounding volume: each node v corresponds to a convex volume that completely contains all the geometry represented by the sub tree rooted at v . Therefore, if some other geometric entity, such as a point, ray, object, etc., is found to not intersect the bounding volume, then no intersection computations need be performed with any geometry within that volume. Consider as an example a situation in which we are given some test point and we want to find which object if any this point lies in. Initially, we know only that the point lies somewhere space ( Figure 20.12). By comparing the location of the point with respect to the first partitioning hyper pl...
Read more

Drawing Trees:HV-Layout

By -
Image
HV-Layout An hv -layout of a binary tree is an upward (but not strictly upward) straight line orthogonal drawing with edges drawn as rightward horizontal or downward vertical segments. The ”hv” stands for horizontal-vertical. For any vertex v in a binary tree T we either have: 1. A child of v is either • aligned horizontally with v and to the right of v or • vertically aligned below v . 2. The smallest rectangles that cover the area of the subtrees of the children of v in the layout do not intersect. Figure 45.12 shows an example of a hv -layout A hv -layout is generated by applying a dived and conquer approach. The divide step constructs the hv -layout of the left and the right subtrees, while the conquer step either performs a ho rizontal or a ver t i c a l combination . Figure 45.13 shows the two types of com- bination. If the left subtree a node v is placed to the left in a horizontal combination and below in a vertical combination, the layout preserves ...
Read more