An Eulerian Circuit is a closed path in a graph that visits every edge exactly once and returns to the starting vertex. A connected graph has an Eulerian Circuit if and only if every vertex has an even degree.
An Eulerian Path is a trail in a graph that visits every edge exactly once, and it exists if and only if the graph is connected and has exactly zero or two vertices of odd degree. If a graph has exactly two vertices of odd degree, any Eulerian Path will start at one of these vertices and end at the other.
Graph connectivity is a fundamental property that determines whether there is a path between any two vertices in a graph, indicating how well the graph is 'held together'. Understanding connectivity is crucial for analyzing network robustness, optimizing routes, and ensuring efficient communication in various applications like computer networks and transportation systems.
Path and cycle decomposition involves the partitioning of a graph's edges into disjoint paths and cycles. This decomposition is a fundamental technique used to analyze and solve problems related to graph structure, connectivity, and optimization.