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**Graph** - In MA010 graph is a pair $G = (V, E)$ s.t. $E \subseteq {V \choose 2}$.

**Complete graph 

**Subgraph** - $H$ is subgraph of $G$ iff $V(H) \subseteq V(G)$ and $E(H) \subseteq V(H) \cap {V(G) \choose 2}$.

**Subdivision** - $H$ is a subdivision of $G$ iff it can be obtained by edge $subdivision$ meaning adding vertex $w$ and replacing $uv$ by $uw$ and $wv$ (obtained by replacing subset of edges by paths).

**Minor** - $H$ is a minor of $G$ iff it can be obtained from $G$ by 

1. vertex deletion (on $v$ $\to$ $V' = V / {v}$ and ${E' = \{ e | e \in E \land v \notin e \} }$)

2. edge deletion (on $e$ $\to$ $E' = E / {e}$)

3. edge contraction (on $u, v$ $\to$ $V' = V / \{u, v\} \cup {z}$ and ${E' = \{ e | e \in E \land v, u \notin e\} \cup \{\{z, w\} | w \in n_G(u) \cup n_G(v) \}}$)

1. vertex deletion (on $v$ $\to$ $V' = V \setminus {v}$ and ${E' = \{ e | e \in E \land v \notin e \} }$)

2. edge deletion (on $e$ $\to$ $E' = E \setminus {e}$)

3. edge contraction (on $u, v$ $\to$ $V' = (V \setminus \{u, v\}) \cup {z}$ and \

${E' = \{ e | e \in E \land v, u \notin e\} \cup \{\{z, w\} | w \in n_G(u) \cup n_G(v) \}}$)

**Isomorphism** - Bijection $\varphi: V(G) \to V(H)$ between $G$ and $H$ is an isomorphism iff $xy \in E(G) \iff \varphi(x)\varphi(y) \in E(H)$.

**Dual-graph** - The dual graph $G*$ of a plane graph $G$ is a graph in which: the vertices of $G*$ one-to-one correspond to the faces of $G$ and two vertices of $G*$ are adjacent if the corresponding faces share an edge.

**Clique-number $\omega(G)$** - vertex size of the largest subgraph $H$ s.t. $H$ is a complete graph.

**Chord** - an edge $uv$ s.t. $u$ and $v$ belong to a cycle but $uv$ does not. 

**Bigon** - pair of edges $(u, v)$ and $(v, u)$.

**Head** - in oriented edge $uv$ ($u$ to $v$) vertex $v$ is called _head_.

**Tail** - in oriented edge $uv$ ($u$ to $v$) vertex $u$ is called _tail_.

# Planar graph 

**Planar graph** - A graph $G$ is planar iff there exists a drawing in the plane $\mathbb{R}$ without edge crossings.

**Planar graph** - A graph $G$ is planar iff there exists a drawing in the plane $\mathbb{R}^2$ without edge crossings. Meaning there exists a function $\varphi : V(G) \to \mathbb{R}^2$ and function $\varphi_{uv} : [0, 1] \to \mathbb{R}^2$ for every $uv \in E(G)$ s.t. $\varphi_{uv}$ is injective, continous. The $\varphi(V(G))$ and all $\varphi_{uv}((0, 1))$ are pairwise disjuncitve.

**Plane graph** - A planar graph together with a plane embedding.

**Facial walk** - minimal face bounding walk.

**Dual-graph** - The dual graph $G*$ of a plane graph $G$ is a graph in which: the vertices of $G*$ one-to-one correspond to the faces of $G$ and two vertices of $G*$ are adjacent if the corresponding faces share an edge.

## Theorems, Lemmas and Facts

**Fact** - If $G$ is connected plane graph, then every face is bounded by a closed walk. Minimal such walk is called a facial walk.