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uni/year2/semester1/logseq-stuff/pages/Definitions & Planar Graphs.md

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- #[[MA284 - Discrete Mathematics]]
- **Previous Topic:** [[Introduction to Graph Theory]]
- **Next Topic:** [[Convex Polyhedra]]
- **Relevant Slides:** ![MA284-Week08.pdf](../assets/MA284-Week08_1666785726176_0.pdf)
-
- # Definitions
- What is a **walk**? #card
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- A **walk** is a sequence of vertices such that consecutive vertices are adjacent.
- What is a **trail**? #card
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- A **trail** is a walk in which no edge is repeated.
- What is a **path**? #card
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- A **path** is a trail in which no vertex is repeated, except possibly the first & last.
- The path on $n$ vertices is denoted $P_n$.
- ## Example
- <img src="https://mermaid.ink/img/ICBmbG93Y2hhcnQgTFIKQigoQikpIC0tLSBBKChBKSkKQiAtLS0gQygoQykpCkMgLS0tIEUoKEUpKQpFIC0tLSBEKChEKSkKQiAtLS0gRSAtLS0gQQoK" />
{{renderer :mermaid_uiukfrr}}
- ```mermaid
flowchart LR
B((B)) --- A((A))
B --- C((C))
C --- E((E))
E --- D((D))
B --- E --- A
```
- $(a,b,c,e,d)$ is a **walk**.
- So too is $(a,b,e,a,b,c)$ (not a trail or a path).
-
- What is the **length** of a path? #card
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- The **length** of a path is the number of edges in the sequence.
- ## Cycles & Circuits
- What is a **cycle**? #card
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- A **cycle** is a path that begins & ends at the same vertex, but no other vertex is repeated.
- A cycle on $n$ vertices is denoted $C_n$.
- What is a **circuit**? #card
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- A **circuit** is a path that begins & ends at the same vertex, and no edge is repeated.
- What does it mean if a graph is **connected**? #card
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- A graph is **connected** if there is a path between every pair of vertices.
- What is the **degree** of a vertex? #card
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- The **degree** of a vertex is the number of edges emanating from it.
- If $v$ is a vertex, we denote its degree as $d(v)$.
- ## Handshaking Lemma
- If we know the degree of every vertex in the graph, then we know the number of edges. This is the **Handshaking Lemma**.
- What is the **Handshaking Lemma**? #card
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- In any graph, the sum of the degrees of vertices in the graph, is always twice the number of edges.
- $$\sum_{v \in V} d(v) = 2|E|$$
- # Types of Graphs
- What is a **Complete** Graph? #card
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- A graph is **complete** if every pair of vertices is adjacent.
- This family of graphs is very important.
- Complete graphs are denoted $K_n$ - the complete graph on $n$ vertices.
- What is a **Bipartite Graph**? #card
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- A graph is **bipartite** if it is possible to partition the vertex set, $V$, into two disjoint sets, $V_1$ & $V_2$, such that there are no edges between any two vertices in the same set.
- What is a **Complete Bipartite** graph? #card
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- If a bipartite graph is such that *every* vertex in $V_1$ is connected to *every* vertex in $V_2$ (and vice versa), the graph is a **complete bipartite graph**.
- If $|V_1| = m$ and $|V_2| = n$, we denote it $K_{m,n}$.
- What is a **subgraph**? #card
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- We say that $G_1 = (V_1, E_1)$ is a **subgraph** of $G_2 = (V_2, E_2)$ provided $V_1 \subset V_2$ and $E_1 \subset E_2$.
- What is an **induced subgraph**? #card
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- We say that $G_1(V_1, E_1)$ is an **induced subgraph** of $G_2 = (V_2, E_2)$ provided that $V_1 \subset V_2$ and $E_2$ contains **all** edges of $E_1$ which join edges in $V_1$.
-
- # Planar Graphs
- What is a **planar graph**? #card
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- If you can sketch a graph such that none of its edges cross, it is a **planar graph**.
- What is a **face**? #card
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- When a planar graph is drawn without edges crossing, the edges & vertices of the graph divide the plane into regions called **faces**.
- The number of faces does not change no matter how you draw the graph, as long as no edges cross.
-
- ## Example
- The graph $K_{2,3}$ is **planar**.
background-color:: red
- ![image.png](../assets/image_1666951300835_0.png)
- [[draws/2022-10-28-11-04-05.excalidraw]]
- The planar representation $K_{2,3}$ has **3 faces** (the "outside" region counts as a face).
- Give a planar representation of $K_4$, and count how many faces it has.
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- [[draws/2022-10-28-11-22-12.excalidraw]]
- Why "face"?
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- [[draws/2022-10-28-11-25-20.excalidraw]]
- # Euler's Formula for Planar Graphs #card
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- For any ^^(connected) planar graph^^ with $v$ vertices, $e$ edges, and $f$ faces, we have:
- $$v - e + f = 2$$
- ## Outline of Proof
- Start with $P_2$.
- Here, $v=2$, $e = 1$, $f=1$. So $v-e+f=2$.
- Any other graph can be made by adding vertices & edges (or just edges) to $P_2$.
- Suppose $v-e+f=2$ for a graph.
- If we add a new edge *with* a new vertex, then no new face is created, so $v-e+f$ does not change.
- If we add a new edge *without* a new vertex, then $f$ will increase by 1, so again, $v-e+f$ does not change.
- ## Example
- Is it possible for a connected planar graph to have 5 vertices, 7 edges, and 3 faces? Explain.
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- No. Euler's formula tells us that $v-e+f=2$.
- Here, $v=5$, $e=7$, $f=3$, so $v-e+f=1$.
- Any such graph is **not planar**.
-
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