Question

In a graph, the edges are pairs of vertices, with no ordering - unlike our formalization with relations as sets of ordered pairs. So, here treat the edges as ordered pairs as follows: For any graph \(G=(V, E),\) let \(\hat{E}\) be the set of all ordered pairs \(\left(v_{1}, v_{2}\right)\) and \(\left(v_{2}, v_{1}\right)\) where \(\left(v_{1}, v_{2}\right) \in E .\) Show that \(C O N N\) is the reflexive and transitive closure of \(\hat{E}\).

Short answer

CONN is the reflexive and transitive closure of \( \hat{E} \).

Step-by-step solution

Understanding the Elements

We start with a graph \( G = (V, E) \), where \( E \) consists of edges which are unordered pairs of vertices. \( \hat{E} \) is constructed by considering each edge \( (v_1, v_2) \in E \) as two ordered pairs: \( (v_1, v_2) \) and \( (v_2, v_1) \).

Defining Reflexivity

To make a relation reflexive, we include a pair \( (v, v) \) for every vertex \( v \in V \). Since \( \hat{E} \) doesn't include these pairs originally, we need to add \( (v, v) \) for all \( v \in V \) to \( \hat{E} \) to ensure reflexivity.

Defining Transitivity

A relation is transitive if whenever it includes \( (a, b) \) and \( (b, c) \), it also contains \( (a, c) \). \( \hat{E} \) as defined may not be transitive. We continuously add pairs by taking combinations of existing pairs until for all \( (v_1, v_2) \) and \( (v_2, v_3) \) in \( \hat{E} \), the pair \( (v_1, v_3) \) is also included.

Showing Reflexive and Transitive Closure as CONN

CONN is the relation we obtain when \( \hat{E} \) is both reflexive and transitive for the entire set of vertices. It ensures connectivity between any pair of vertices through a path. Therefore, after adding necessary pairs to \( \hat{E} \) for reflexivity and transitivity, we achieve the reflexive and transitive closure of \( \hat{E} \), represented as CONN.

Key concepts

Reflexive Closure

In graph theory, the concept of reflexive closure is essential when analyzing relationships between vertices in a graph. To comprehend this, it's important to first understand reflexivity. A relation is reflexive if it includes the pair \((v, v)\) for every vertex \(v\) in the set. This means each vertex has a loop back to itself.
For instance, suppose you have a set of elements consisting of three vertices \(\{1, 2, 3\}\). To make this relation reflexive, you must ensure that \((1, 1)\), \((2, 2)\), and \((3, 3)\) are present in the set of pairs. If these pairs are not originally part of your vertex set, you need to add them to achieve reflexivity.
This addition equates to constructing the reflexive closure in the graph. By ensuring every vertex is linked back to itself, you create a complete self-loop structure for all vertices.

Transitive Closure

Transitive closure is crucial in understanding pathways in a graph. A relation is transitive if, whenever it includes the pairs \((a, b)\) and \((b, c)\), it also contains \((a, c)\). This ensures that if there is a path from \(a\) to \(b\) and from \(b\) to \(c\), then there is a direct path from \(a\) to \(c\).
In graph terms, this implies continually examining and extending pathways. Let's say your graph includes edges \((1, 2)\), \((2, 3)\). For the graph to be transitive, \((1, 3)\) must also be added if it does not already exist.
The transitive closure is reached once all such necessary connections are made within the graph, ensuring complete path connections between all reachable pairs of vertices. It is fundamental for establishing comprehensive connectivity and understanding the flow of information or movement across the graph.

Unordered Pairs

Graphs typically consist of edges, which are unordered pairs of vertices. This is distinct because there is no specified direction of connection between the vertices in an edge. For instance, if there is a connection between vertices 1 and 2, it does not matter whether we write it as \((1, 2)\) or \((2, 1)\); both represent the same edge.
When dealing with ordered pairs, we introduce a directionality that can change the interpretation of the relationships. To handle this for practical graph applications, each unordered edge \((v_1, v_2)\) is treated as two ordered pairs, \((v_1, v_2)\) and \((v_2, v_1)\). This transformation into ordered pairs is important for applying concepts like reflexive and transitive closure since they work based on ordered relationships.
Understanding unordered pairs and how they can be expressed as ordered pairs is key for effectively implementing mathematical concepts within graph theory.

Connectivity in Graphs

Connectivity in graphs pertains to the idea of paths that link vertices. It answers the question: can we travel from one vertex to any other vertex in the graph without breaking the path? This concept is vital for understanding how well-connected or integrated the graph is.
Ensuring connectivity involves forming a relationship, often termed as \(CONN\), which guarantees that any two vertices in the graph are connected either directly or indirectly through a sequence of edges. To achieve this, both reflexive and transitive closures are essential.
Creating a reflexive and transitive closure, essentially transforms the set of ordered pairs into a comprehensive connected network. This process ensures every vertex can reach every other vertex via possible pathways, thus establishing full connectivity in the graph.

• Reflexivity ensures every vertex connects to itself.
• Transitivity extends the paths to connect indirect vertices.
• These create \(CONN\) that defines a fully connected graph.

By comprehending these processes, the foundation of connectivity in graph theory becomes clear, enabling further exploration of various complex graph structures and their applications.