## What is dual graph in graph theory?

In the mathematical discipline of graph theory, the dual graph of a plane graph G is a graph that has a vertex for each face of G. The dual graph has an edge for each pair of faces in G that are separated from each other by an edge, and a self-loop when the same face appears on both sides of an edge.

Is the dual of a planar graph planar?

A dual graph is defined for planar graphs, which do not necessarily have to be simple graphs, i.e. the original graph can contain loops and multiple edges. Loops or multiple edges in the original graphs transform into edges in the dual graph and vice versa. Thus, both graphs have exactly the same number of edges.

### What does it mean to be 2 connected?

2-Connected Graphs. Definition 1. A graph is connected if for any two vertices x, y ∈ V (G), there is a path whose endpoints are x and y. A connected graph G is called 2-connected, if for every vertex x ∈ V (G), G − x is connected.

What makes a Euler circuit?

An Euler circuit is a circuit that uses every edge of a graph exactly once. ▶ An Euler path starts and ends at different vertices. ▶ An Euler circuit starts and ends at the same vertex.

## How is dual graph formed from its planar?

The planar dual of an embedded planar graph G is the graph G formed by placing a vertex inside each face of G, and connecting those vertices of G whose corresponding faces in G share an edge. Each vertex in G has a corresponding primal face and each edge in G has a corresponding primal edge in the original graph G.

What is combinatorial dual?

Then a graph is a combinatorial dual of if there is a one-to-one correspondence between their sets of lines such that for any choice and of corresponding subsets of lines, where is the subgraph of with the line set .

### What is a 3 connected graph?

A graph G is 3-connected provided between any two vertices x and y there are three paths that meet only at x and y.

What is a 2 edge connected graph?

A connected graph is 2–edge connected if it remains connected whenever any edges are removed. A bridge (or cut arc) is an edge of a graph whose deletion increases its number of connected components, i.e., an edge whose removal disconnects the graph. So if any such bridge exists, the graph is not 2–edge connected.

## What is difference between Euler circuit and Euler path?

An Euler path is a path that uses every edge of a graph exactly once. An Euler circuit is a circuit that uses every edge of a graph exactly once. ▶ An Euler path starts and ends at different vertices. ▶ An Euler circuit starts and ends at the same vertex.

What is the difference between an Euler circuit and a Hamilton circuit?

Important: An Eulerian circuit traverses every edge in a graph exactly once, but may repeat vertices, while a Hamiltonian circuit visits each vertex in a graph exactly once but may repeat edges.

### What is the difference between simple graph and multigraph?

A graph is defined to be a simple graph if there is at most one edge connecting any pair of vertices and an edge does not loop to connect a vertex to itself. When multiple edges are allowed between any pair of vertices, the graph is called a multigraph.

Is dual graph connected?

If we follow the line from xF to xE, we “describe a path” in the dual graph from F to the external face. Thus, each vertex of the dual graph is connected to the vertex corresponding to the external face, which means that the dual graph must be connected.

## What is a connected planar graph?

A planar connected graph is a graph which is both planar and connected. The numbers of planar connected graphs with. , 2, nodes are 1, 1, 2, 6, 20, 99, 646, 5974, 71885, (OEIS A003094; Steinbach 1990, p.

What is a combinatorial graph?

Combinatorally, graphs are just a set of objects (the vertex set) and a set of equivalence relations (the edge set) regarding the arrangement of the objects. For example, a triangle is a graph with three vertices and three edges. So the vertex set may be (x,y,z) and the edge set (xy,yz,zx).