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The bipartite realization problem involves finding a bipartite graph that matches a given set of constraints or properties, specifically with respect to a prescribed set of edge weights or degrees. In a bipartite graph, the vertices can be divided into two disjoint sets such that no two graph vertices within the same set are adjacent.

The Canadian Traveller Problem (CTP) is a combinatorial optimization problem that extends the classic Travelling Salesman Problem (TSP). It arises in scenarios where a traveller must visit a set of locations (cities or nodes) while adhering to certain constraints.

The Chinese Postman Problem (CPP), also known as the Route Inspection Problem, is a classic problem in graph theory. It involves finding the shortest path or circuit that traverses every edge of a given graph at least once. The goal is to minimize the total distance traveled, effectively allowing the "postman" to deliver mail along the edges of the graph without unnecessary repetition.

In graph theory, a **clique cover** of a graph is a partition of the vertex set into cliques. A **clique** is a subset of vertices that forms a complete subgraph, meaning every pair of vertices within the subset is connected by an edge. Therefore, a clique cover is a way to divide the graph's vertices into groups where each group is a clique.

The Clique problem is a well-known problem in graph theory and computer science, particularly within the field of computational complexity. A clique in a graph is defined as a subset of vertices such that every two distinct vertices in the subset are adjacent, which means there is an edge connecting every pair of vertices in that subset.

A **Connected Dominating Set (CDS)** is a concept from graph theory, particularly in the study of network design and communication networks. It consists of a subset of vertices (nodes) in a graph that satisfies two main properties: 1. **Dominating Set**: The subset of vertices \( S \) is a dominating set, which means that every vertex not in \( S \) is adjacent to at least one vertex in \( S \).

Correlation clustering is a type of clustering algorithm used to group a set of objects based on the correlations among the objects rather than traditional distance measures. Unlike typical clustering methods, which often rely on distance metrics (like Euclidean distance), correlation clustering focuses on maximizing the number of pairs of similar items within the same cluster while minimizing the pairs of dissimilar items in the same cluster.

The Degree-Diameter Problem (DDP) is a classic problem in the field of graph theory and combinatorics. It focuses on the trade-off between the degree of vertices in a graph and its diameter. Specifically, the problem seeks to determine the maximum number of vertices \( N \) in a graph given two constraints: the maximum degree \( D \) of any vertex and the maximum diameter \( h \) of the graph.

The Deterministic Rendezvous Problem is a classic problem in distributed computing and algorithm design, particularly in the fields of multi-agent systems and robotics. The problem involves two or more agents (or entities) that must meet at a common point (the rendezvous point) in a distributed environment, without the use of randomization. ### Key Characteristics of the Problem: 1. **Determinism**: - The behavior of the agents is predetermined and follows a specific set of rules or algorithms.

The Digraph Realization Problem is a key issue in graph theory, specifically within the context of directed graphs (digraphs). The problem can be described as follows: Given a set of vertices and a collection of directed edges (or arcs), the goal is to determine whether there exists a directed graph (digraph) that can represent those edges while satisfying specific combinatorial properties.

The domatic number of a graph is a concept in graph theory that describes the maximum number of disjoint dominating sets that can be formed within that graph. A dominating set is a subset of the vertices of a graph such that every vertex not in the set is adjacent to at least one vertex in the set.

In graph theory, a **dominating set** for a graph \( G = (V, E) \) is a subset \( D \subseteq V \) of the vertices such that every vertex not in \( D \) is adjacent to at least one vertex in \( D \).

In graph theory, an **edge cover** of a graph is a set of edges such that every vertex of the graph is incident to at least one edge in the set. In other words, an edge cover is a collection of edges that "covers" all vertices in the graph.

An **edge dominating set** in a graph is a subset of edges with the property that every edge in the graph is either included in the subset or is adjacent to at least one edge in the subset.

A **Feedback Arc Set** (FAS) is a concept in graph theory that refers to a specific type of subset of edges in a directed graph (digraph). The purpose of a feedback arc set is to eliminate cycles in the graph. More formally, a feedback arc set of a directed graph is a set of edges such that, when these edges are removed, the resulting graph becomes acyclic (i.e., it contains no cycles).

A **Feedback Vertex Set (FVS)** in a graph is a set of vertices whose removal makes the graph acyclic, meaning that it eliminates all cycles in the graph. In other words, a feedback vertex set is a subset of vertices such that when these vertices are removed from the graph, the resultant graph contains no cycles.

Frequent subtree mining is a data mining technique that focuses on identifying substructures (or subtrees) that appear frequently within a collection of tree-structured data. This process is particularly important in domains where data can be naturally represented as trees, such as in biological data (e.g., phylogenetic trees), XML data, and other hierarchical structures. ### Key Concepts 1.

A **good spanning tree** is not a standard term in graph theory, but it can be interpreted in a few different ways depending on the context. Generally, a spanning tree is a subset of a graph that includes all the vertices and is a tree structure without any cycles.

Graph coloring is a concept in graph theory that involves assigning labels, or "colors," to the vertices (or sometimes edges) of a graph under certain constraints. The primary goal is to ensure that adjacent vertices (or edges) do not share the same color. This is useful in various applications such as scheduling, register allocation in compilers, and frequency assignment in telecommunications.

Graph cut optimization is a technique used in computer vision, image segmentation, and machine learning to partition a graph into distinct parts. The method involves modeling data as a graph, where nodes represent pixels (or superpixels) and edges represent relationships (or similarities) between these nodes.