Trees and Graph Algorithms

- Euler Path: A path in a graph that visits every edge exactly once. - Euler Circuit: A closed path (circuit) in a graph that visits every edge exactly once and starts and ends at the same vertex. --- A connected graph G has an Euler Path if and only if it ...

- Numbers for GCD computation: $a = 12$, $b = 18$ - Required: necessary and sufficient conditions for Euler path and Euler circuit; GCD via Extended Euclidean Algorithm expressing $\gcd$ as a linear combination. No data missing. --- - Euler Path: A trail th...

An Euler path (also called an Eulerian path) is a path in a graph that visits every edge exactly once. If such a path starts and ends at the same vertex, it is called an Euler circuit (or Eulerian circuit). - A connected graph has an Euler path if and only ...

Problem: Find shortest path from $a$ to $z$ using Dijkstra's algorithm. Critical issue: The question refers to a graph ("following graph"), but no graph image, edge list, or weight data is provided in the text supplied to me. Dijkstra's algorithm requires: ...

Dijkstra's algorithm is a greedy algorithm that finds the shortest path from a source node to all other nodes (or a specific destination node) in a weighted graph with non-negative edge weights. --- --- --- Consider the graph: Find shortest path from A to E...

Dijkstra's algorithm finds the shortest path from a source node to all other nodes (or a specific goal node) in a weighted graph with non-negative edge weights. --- - dist[] : array storing shortest distance from source to each node - visited[] : boolean ar...

- Algorithm required: Dijkstra's algorithm - Source vertex: $a$ - Destination vertex: $z$ - Graph: "the given weighted graph" - the actual graph image/edge weights are NOT provided in the question text. Missing data: The specific weighted graph (vertices, e...

A spanning tree of a connected, undirected graph G = (V, E) is a subgraph that: - Includes all the vertices of the graph - Is a tree (connected and acyclic) - Has exactly V - 1 edges A graph with n vertices will have a spanning tree with exactly n - 1 edges...

A cut vertex (or articulation point) of a connected graph G is a vertex whose removal (along with all edges incident to it) increases the number of connected components of the graph. In other words, vertex v is a cut vertex if G is connected but G - v is di...

Tree traversal means visiting every node of a tree exactly once in a systematic way. The three standard depth-first traversal methods differ in when the root (parent) node is visited relative to its subtrees. --- --- Algorithm: 1. Visit the root node 2. Rec...

Tree traversal is the process of visiting each node in a tree data structure exactly once in a systematic and well-defined order. It is used to process, search, or display all the nodes of a tree. Unlike linear data structures (arrays, linked lists), trees ...

The question refers to "the given graph," but no graph (vertices, edges and weights) is reproduced with the question text available here. The working below therefore uses the standard 9-vertex textbook MST example so that the method is demonstrated in full....

Kruskal's algorithm is a greedy algorithm used to find the Minimum Spanning Tree (MST) of a connected, undirected, weighted graph. It builds the MST by always picking the smallest available edge that does not form a cycle. --- 1. Sort all edges in non-decre...

A spanning tree of a connected, undirected graph G = (V, E) is a subgraph that: - Includes all vertices of the graph - Is a tree (connected and acyclic) - Has exactly V - 1 edges A graph with n vertices will have a spanning tree with exactly n - 1 edges. --...